A Stereovision-based navigation system for a mobile robot

HAL Id: inria-00074776

https://hal.inria.fr/inria-00074776

Submitted on 24 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A Stereovision-based navigation system for a mobile robot

Michel Buffa, Olivier Faugeras, Zhengyou Zhang

To cite this version:

Michel Buffa, Olivier Faugeras, Zhengyou Zhang. A Stereovision-based navigation system for a mobile

robot. [Research Report] RR-1895, INRIA. 1993. �inria-00074776�

!"#$

%'&)(+*,(.-/(01(+*324%5076

89 91:<;59>=?:<;'9

9 %@ %'AB2 =DC 01(+*3E5- 9

9 ( 9 %F-G($2 C 01(+*3E5- 9

HJI7ILK

= 2M-G( 9N8O9 &"P- : *3246 9 &

QRSTRVUJW

I7X

U

IJH

G2JY ; *301Z%J(+*3Y[2M6\*3&

] = 0^%

:_9

tvuLwJU7x

yz<{}|4z+~€@‚ƒ

„†…1‡€…_ˆŠ‰\‹€Œ1 Ž€B‘$’€$ˆ†“O‰,”•‰\… –

— ˜r™sš›šcœcžJ˜Ÿž7œ¡  ¢£¤— ˜Ÿš¥

  —g¦ž7§ —N™sžJœ¡  ˜Ÿ¨ ˜©™ªš« ¬œ¡› —

« œ¡£¡ž_­®š › œ¡£œ¤™

¯ * :<;59 6[Q]Ÿ]J

6\*,°*

9>=

]J±

² ;'9 %'³µ´^2M- ²P¶±

!" # $&%

' ( ) *

$ #+ , * * # #

.- / 0-

« 13254687!£9;:=< œ>731@?1A6CB ¥ D¬<E9FG6HBI<EJ

K4L68MFGNPOP9 K4<EMLF

QSRUTQAV W5XZY\[\]_^`3VUaIbc]dY8XZed]dfgh ikj

lZm

i

lZn

W5XZY\[\]_^`3VUaIbc]dY8XZed]dfoqpsrtpvuxw\y5z{^|aC}~p

€ƒ‚k„|…‡†xˆ ‰Š|‹‹„Œ|Ž‘I…„x’“…v”|•I…s„–’@‹•E—‹„˜Š|™š›•„‡ŒZŽ˜‘œ…„–’“…v”•I…‡„–’@‹•



‘Z„˜”|™ŒŽ‘5…‡„G’3…”Z•Z…„–’ž‹|•

Ÿ¡ ‡¢3£¥¤‡¦˜§©¨«ª5¦‡¬­¤‡¤¯®3°3°“§©¨¯±«²´³\£¥µ\°S¬¶¨«±‡·@¸t¹º¤«°»¨«£¥±°3¨¯§´¼¡²´½¾±|¿vÀ¶ÁAÂ3Ã

!"$#%&(')*+-,.+0/123#"!* 54$6*789&*82 !6*82:5;*<" !=

4*2&2$2&>@?AB*(CD#E4F(G#HI$2&>"7G9-FG2KJ

L H&BCMG#N4$%G#OOPH$&Q'$*R"$2&& !H49*8B7I$2:S=T"5CU5*V*O#2:&H&

$2WD*;'B**2DX#( **#4$?V!4E$7"$2&>:YZ[C\*]?^5&%2W7G$2&CM2_*8`!*

#"'2ba:*82dcfeXgJh&W"3YZiC\*D !2W!*$2j4E$2&kS>CM2;(!l&;* !m?A5Cn&

$C\*>B*82@ !2"5H !D$2:&G$CM2-FG2JB&=*%H"o?AB*(7*"$"=\8?^;*G")T

$2 54FHG$2&>Q#";* 54$*7&*2 !(*2@;*8<- !5=(4*2&2$2&>J

p&GM&W"@"'q!*")&rs'&<" !(2j&Q>H&2ut(&qYGZ14E$2&f">CM2;R

#&;*$2f*@"'v0G$CM2I$2*4VC\*8KrK'MI$CM4EN?w=f&(C\*8k*G !;G$2&>:"CD6-FCD4F@>v

CMS9 Q !"$"*&rB*2xH":5&:C\*yF2$2&>dzZ{S>CM2;(| !G2"H !*3S!4N4*SFG2Kr

CM@& 5"!4$=}*QS9*82&>H4*"$2Kr?~5&>H&2utJM5"!4E49*8"$2d*D"7;*4

*7y*82W;*8>€



+0*&*&@5&"5H !H&?‚&27$2&CD2W~-F2 !*y4N4‚"5M">CM2;*

>B8?‚"*2&>4$]$2:&R"!4E4*"$2Kr



+0~ *2#%!ƒo 5$2WS4F=: !GCM&H&3„I&_*y4F>G"$&C…'H"%?w&"*2&>H4*"$2Q*

*M !CM4$!lF"=?V†(„‡4FG>`ˆŠ‰VH&‹&*8rE?`ˆŒ9X&%2H&C(#?‚8F2;XH"‰;r



+0B=2*C( r$2:5&"2"%*B">GCM2W!` *82#*&@B"H&#&5;* !D?wC…*2

!lW""$2&>@"*2&>H4*SFG2:!ƒo 5$2"4$=r

fBHS`59Ž"*2&>H4*SFG2\*V*%SH&&?w?wH&&s&& !I$2&>JŽf;"V5C($2&

?A5"* !rI$CM4$=#=MC\*5)$2&>M&"%"*2&>4$'X9 5q*8~CD&-=r*>W*$2*(7=\-FCD4F

&& !I$2&>r‚*82:5&2qHS5&>G;*&?wCD:#=:5&W"5"*2&>4$%@>2&;*8 !4E4EI$2

?A5B;*8<- !G"$JK'B&2D2&'‘S2"=(&*;*%** PH$D&B>GH&2RtVC@*R9O*-N4$=

H&‹&*5MHI$2&>M5&~2 ! !CM&H&!*"$2*4’&5"$V8?“5&_Z!4*H&2*=MS9*82&>H4*"$2:*2

&%&& !]9`F5;*’J

f%"&'d!l*CM4$Ž$2\'B 5H&V5#O2*7$>W*5?w!4$=($2:*%*4$2W!V27GF52&v

CM2BHI$2&>\59X"=TSC:J

(;*& !SF4$M5;*7y*E4VK !H(*4$(4*#;*5$M.#G7R(4+-,.+0/i"H&4*

4*2E‹ *"$2q;*8<- !5$BO47G$CM2]#";* 54$OGH&`H&25#OCM#E4$r* 4*

4*"y7GI$2KJ,B5B#sCM#E4$"PH$BH&2@"=TSCMBB7GI$2@"$2&T !H4*$

*;'B*V *8*#4$`` !4EF7s‚ *8KYZs427GF52&2&CM2*H&2&s?wPGH&2 !V*4E4*2

qa*QceXgJ R *RYZ…"\ !GCMWS2W(o">GCM2W!%o8F5PGHXS2W *y49 !H4

**"$:oFC@*>5 !2"H$;@2 G$CM2I$2J%,XHD4$H&"E4E"23H&

* !CM4E$GF7G;"(*G &rs !CDCM\4*2E@H&2&:!*<" !$M7$R4$(#S;* 54$J

H&V !VHl2Ž* 5&r2&Hs&5T 2sB4* C\*2B"H$7*2W5€4$O">CM2;

YZ["2`&5<" s"H&]4FS4“*2:#&2$BH&2& *5~#G$CM2I$2&2&!4$4$"!K !% *"

2"H$I$CM4EN#$*o4*o@ !"$>%C"PGH&~I$CM4$r&HRH&2&M"*2&>GH49*8"$2

:Z!4*H&2*=m@">CM2;(zZ "\*4$!K !HJ

S9*82&>H4*"$2d*34$HI$H&;

*7y*2;*> €

'&

4E4$\"*G&*&(*D4*"H !H&56642W7G$2&2&CM2&HPGH&\5H4$R">CM2;

"2~B*S9*82&>4$J

'&

4E4$H&)~ *4 !H4%5Ž!ƒo * !CM2„w4*4$>"$&CD_PGH&%2&HOH&"E4E9S2B*RH&2&

!CM4$!lW$(†(„w4$>ˆŠ‰^*8`C69S**B<"H&rI‚ˆQ"O4F%2&GC#&5_%$2W;)*5"*2&>H4$J

'&

4E4$qS6=2*C(PGH&f&*2R4*3CM"H&5+Hd *GPH&3">CM2(H&,*<"H&MH

"$H&2&%"*2&>H4*"$2Q!lW"!*2W5%;*CM2J

"*2&>GH49*8"$2\S`H&SN4E" !CMCD_"H&&GH&]*H&O;*yF5CM2W!JW,XH-v

C($2&2ŽH&*#G;R4"* !X4E$#&_-FCD4FCD2W2@C\*;PGH*24FOS9*82&>4$`PGH“S2W

7r’&H H&"E4N"G2%4$(>;*8&&?wC%* !S9*82&>4$H&%>%2.XH&2&M;*8<- !8F5

*2B !84N4EI$2KJ

*PGH&@?A89(PGH&:(2&H&7!4E4$@2&2.R&72*2MM *&H&!7GI$2}S2WMG"v

2$#4F r&4*M *5_zGZh";*89CD2WBC69S**B<"H&B>G;* !%*Hl\&&S/^$2W*82WO

4*("*2&>H4*SFG2qZ!4*H&2*=J

,BGHCM220*D4*@2|M !M;*8&!l&CM4$R!4%(2&55#%!l&4$;*2

H&2&M !M2 !GC#&rK*7 (* *GPH&;*%4* !G2"H !"$2(4"* !(4E$#&@4*

4$T *4E*SFG2k]#";* 54$*H&H&!`HQ#J

!

"

1$#&% ')(*+-,$.0/2143151,76984884884848848848:8;8:84884848848848848:8;8:848;8:848 <

1$#N” =?>@.BAC1DE+@F&3GH+@(1*1JILKM15GBNO13P+QKR.B3S+@(14GB>T.BU3, 84884848848848848:8;8:848;8:848 V

1$#1 WYXZ,*+@F[3!G\+T(1\”BI]N^*_XO/`F&+@(F[3P+@1GJ>Q*+@F[3!GH+@(1431/aKM1+`.b?X>T.BAC1DE+T1,KM1GJNO13P+QKc8 d

1$#1$#&% ef1X!>@1KC13$+g*_+TF[.B3.bRh[F&314KM15GBNO13P+QKi8;8:84884848848848848:8;8:848;8:848 d

1$#1$#N” j71>@GJF[3GH+T(1\”BILKC1GBNk13$+gKEl!KMF[NkXh[FnmZD*_+TF[.J3\.b?+T(1\”BI]N^*_Xi8;8:848;8:848o%p

1$#1$#1 =q>T1KM15>@rPF[3GH+T(14r$FsKMF[t!F[h[F&+Cu0*_+@+T>@F&tU+@1v8:84884848848848848:8;8:848;8:848o%1

w x:

y"

{z

|

- $

}~Y

~C

y!

P

#&% €R.B3KM+@>TUDE+TF[3Gk*\I1h‚*U3*uO+@>TF‚*3GBUhs*_+TF[.B3S._bq*Q”JI]N*Xƒ848848848:8;8:848;8:848o%1

#N” €R.B3KM+@>g*_F&31,7I1h‚*U3*uO+@>TF‚*3GBUhs*_+TF[.B39848:8;8:84884848848848848:8;8:848;8:848o%<

#1 WYXZ,*+@F[3!GO,$uP3*NOF‚D*h[h&u0*_3715„$FsKM+@F&3G+T>@Fs*_3GJUh‚*+@F[.J3 848848848848:8;8:848;8:848o%…

#† j‡*_>Tˆ$F&3GH1NOX!+MuH+T>@Fs*_3GJh[1K‰lDE.BNOX!U+@F&3G+T(1-b>@11:KMX*BDE1 848848848:8;8:848;8:848o%d

‹ŠP

Œ"

|

Z

<$#&% Ž3$+T>@.P,$UDE+@F&.B3848:8;8:84884884848848848:8;8:84884848848848848:8;8:848;8:848n”$%

<$#N” ‘Y./’/21:DE.BNkXU+@1“*”X*+@( 884848848848:8;8:84884848848848848:8;8:848;8:848n”"1

<$#1 •‹F&NOXh[Fnbu$F&3G4+@(14X*+@( 884884848848848:8;8:84884848848848848:8;8:848;8:848n”B–

— ˜

||

|™

š !›

œ

|‚

!›

 x:

| !›

› ž y‹Ÿ0|

"

š

­ 13J O ¬ 1@FP9LBI68J

% Y>gD (F&+@1D‰+@U>T1Y.bR.BU>KCuKC+@1N 84848848848:8;8:84884848848848848:8;8:848;8:848 

” 2(1:N*_XON^*_ˆPF[3G:X>@.PDE1K@K*_>QDQ(F&+@1DE+TU>@1c8:8;8:84884848848848848:8;8:848;8:848 –

1 +Cu$XFsD*_h F[3,P.‹.B>kK@DE13!17*3, +@(1X>T.BAC1DE+TF[.J3.b-+T(1 1BI KM1GJNO13P+QK;X!>@.P,$UD‰1,

b>@.JN *HKM+@15>@1.HrPF[1/’._b +@(!F‚K`K@D‰131 848848:8;8:84884848848848848:8;8:848;8:848 V

 ?h[h&F[XKC1K2._b U3DE15>@+Q*F[3P+Cu”b.J>Y+T(14NOF‚,$X.JF[3P+QK?._bR+T(1\”BIiKM1GJNO13P+QK .bymGBU>T1(1 848 …

< yF&>QKM+KC+@1XRKMF[NkXh[FnmZD*_+TF[.J3\.bR+@(1:KC1+`._b KC1GBNk13$+gK?._b?mGJU>@1 18848:8;8:848;8:848o%”

– WYXZ,*+@F[3!G\+T(1\”BI]N^*_X# 21Eb.B>T1*3,7*b+@1>`+T(1-bUKMF&.B3 8848848848:8;8:848;8:848o%”

V ')(13 NO1>TGBF[3!G7+C/2.cKC1GBNO153$+gKEl?+@(!17rPF‚KMF&tF[h&F[+Cu7.b41*JD (X*>@+0._b4+T(1>@1KCUh[+TF[3G

KM15GBNO13P+2FsK`DE.BNkXU+@1, 884884848848848:8;8:84884848848848848:8;8:848;8:848o%1

… …”BI9N*_XO.b *0K@DE13!1v884884848848848:8;8:84884848848848848:8;8:848;8:848o%–

d WY3D‰.B3KC+@>Q*F[31,*3,DE.B3KM+@>g*_F&31,IY1hs*_U3*uc+@>TF‚*_3!GBUhs*_+@F&.B3._b +@(!1KM1+H.b4KC1G

NO13P+QKR._bmGBU!>@1H… 8:84884884848848848:8;8:84884848848848848:8;8:848;8:848o%_V

%p ')(13 *\KC1GBNk13$+ F‚KDE.B3KM+@>g*_F&31, l+T(1rPF‚KCF[tF&h[F[+Cu0*+@+@>TF[tU!+@14F‚K`F[3(!1>@F&+@1,\tPu+@(1

*B,!,$1, X.JF[3P+QKY848:8;8:84884884848848848:8;8:84884848848848848:8;8:848;8:848o%_V

%B% WYXZ,*+@F[3!G\+T(1:,$1h‚*U3*uO+T>@F‚*3GBU!h‚*_+TF[.J3 8:8;8:84884848848848848:8;8:848;8:848o%…

%” `*BKCF‚D-X>TF[3DEF&Xh[1 ._b UKMF&3G\+T(14r$FsKMF&tF[h&F[+CuX>T.BX1>T+Muk+@.0N*_>Tˆ\1NkX+CuH+@>@Fs*_3GJh[1K‰#‡848o%d

%1 W-KCF[3GH+T(14r$F&1/`X.BF[3P+QK2+T.\N^*_>@ˆ01NOX!+MuH+T>@Fs*_3GJh[1K‰#84848848848848:8;8:848;8:848n”Bp

%E =y. K@KMF&th[14X*JK@KT*_GB1K-,$.03.B+>@1X!>@1KC13$+ +@(F&3G‹K2+T(*_+(*r‹14t113SKM1153 8;8:848;8:848n””

%< 2(1:1NOX+CuH+@>TF‚*3GBh&1K`D*_3t1F&3$+T1>@X!>@1+T1,\*JKY*HGB>g*_X( # 848848848:8;8:848;8:848n”"1

%– •‹.JNO11„*NOXh&1\.b4X*_+T(KODE.JNOXU+T1,{b>T.BN +@(1GJ>Q*_X!(}b.B>@Nk1, tPu +@(1S1NOX+Cu

+@>TF‚*3GBh&1K 848848:8;8:84884884848848848:8;8:84884848848848848:8;8:848;8:848n”_

%_V •‹.JNO1-X.‹KTKMF[t!h[1X*BK@KT*_GJ1KYN*uH(*r‹143.0/`*uO+@.HGJ.\+T.\+T(1N‡# 8848:8;8:848;8:848n”B<

%… ‘Y./’/21:KMF&NOXh[FnbuH+@(14X*+@( 84848848848:8;8:84884848848848848:8;8:848;8:848n”B–

%d 2(1:tš1KM+KMF&NOXh&Fm1,HX*+@(‡D‰.BNOXU!+@1,”b>T.BN +T(14X.‹K@KCF[th&14X*_+T(Kf.bR+@(!1“mGBU>T1\%5– ” V

”Bp ')(13)3!. X*+@(’DE.JUh‚,t1‡DE.JNOXU+T1, +@(1S>@.Bt.J+ACUKM+kXš15>Mb.B>TNKH* KCN*_h&h`K@*b1

NO.r‹1NO1‰3P+?+@.kKM11:*315/9*_>T1* 848848848:8;8:84884848848848848:8;8:848;8:848n”B…

”$% •‹.JNO1->@1KMUh&+QK\848:8;8:84884884848848848:8;8:84884848848848848:8;8:848;8:848 1$%

”” •‹.JNO1->@1KMUh&+QK\848:8;8:84884884848848848:8;8:84884848848848848:8;8:848;8:848 1”

”"1 •‹.JNO1->@1KMUh&+QK\848:8;8:84884884848848848:8;8:84884848848848848:8;8:848;8:848 1"1

”_ •‹.JNO1->@1KMUh&+QK\848:8;8:84884884848848848:8;8:84884848848848848:8;8:848;8:848 1_

”B< e`1X>@1KM13P+Q*+@F[.J3._b +@(1-b>@151KCX*BD‰1*b+@1>4%y”\rPF[1/K‡84848848848848:8;8:848;8:848 1B<

”B– >@F‚*3GBU!h‚*_+TF[.J3‡*b+T1>4%”HrPF[1/Kƒ848848848:8;8:84884848848848848:8;8:848;8:848 1B–

U>`NO.JtF[h&1->@.Jtš.J+-D5*_37UKM1:*”rPF‚KCF[.J3 N^*BDQ(F[31/`(FsD (O(*BK`tš1513‡,$1KMF[GJ31,*_3, tUF&h[+ /`F[+@(!F[3S*_3

?U>@.JX1*_3 RKCX>@F&+ X!>@.BAC1D‰+l‹X>@.JAC1DE+R=Rd_ p$l!*_h‚KC.HD*h[h&1,HI-j l_b.J> IY1X+T(O*3,Oj7.B+TF[.J3 Y3*_h[u!KMFsKE#

')F[+T(.BU!+-13P+@1>TF[3GOF&3$+T.O+@(!10,$1+Q*F[hsKf.b`+@(!F‚KYN*JD (!F[31‹lF&+-F‚K KMU^DEF[153$++@.KT*u‡(1>@1:+@(*+4F[+4D*3

X>T.PDE1K@K`+@(!>@114F[N^*_GB1K2*BD UF&>@1,Hb>@.BN *+T>@F[3!.PDEUh‚*>-KC+@1>T1.H>@F&G0*_3,SDE.BNkXU+@1 +@(14X.‹KCF[+@F&.B3 *_3,

.B>TF[13P+Q*+@F&.B3F&3+T(>@11 ,$F&NO13KCF[.J3K ._bZ* KM1+._bh&F[31 KC1GBNO153$+gKE# 2(1KM12h&F[31 KC1GBNO153$+gK*_>@1 X>@.P,PUDE1,

tPu X.Bh[uPGB.J3*_h *_X!X>@.„PF[N*+@F[.J3K._b 1,$GJ1KHF[3+@(1+T(>@11F&N*_GJ1KE# 2(1‡KC+@1>T1.c*_h&GB.B>TF[+T(N +@(*_+

N*+QDQ(1K2+@(!1X.Bh&u$GJ.B3*h2DQ(*F[3K`(*BKtš1513c,$1KTDE>@F&tš1, F[3 …‹V *_3,‡F[NkXh[1N01‰3$+T1,”F&3‡X*_>g*_h&h[1h

.B3 *Ot.‹*_>g, /`F[+T( 1j7.J+@.B>T.Bh&h‚*<B–JpBpBp‡I4•P=KE# 2(FsKYt.‹*>Q, l?X*>@+4._b +@(10I j LN*BDQ(F[3!1‹lš(*BK

t113S,$1KCF[GJ31,*_3,tUF&h[+?AC.BF&3$+Th[u0t$ukŽ-eŽ *_3,Sj `e #

f+:+@(1k+@F[Nk1\.b-+T(F‚K4/`>@F[+TF[3G‡+T(1\m3*_h`r‹1>gKMF[.J3 ._b +@(!1ON*BDQ(F&31\FsKfACUKC+0*rB*F[hs*_th&1OF[3 .JU>

h‚*t.B>Q*+@.J>@u‹l*3, / 1-/`F[h&hKM+g*_>T+`+@.HUKC1-F[+`F&3$+T13KMF&r‹1h[u # 2(14+@(>T.BUGJ(XU+`+TF[NO1F‚KRt1+C/2113 %4*_3,

<7‘‹l?,$15Xš153,$F[3!G7UX.B3 +@(1O15„*JDE+4(*_>g,$/`*>@1DE.J3$mGJU>Q*+@F[.J3# 2(FsK;NO1*_3K-+@(*_+lF[3 +T(10tš1KM+

D*JKM1‹lB/ 1`/`F[h&h>@1D‰.B3KC+@>@UDE+R*Y+T(>@11 ,$F&NO13KCF[.J3*_hP/`F[>T1Eb>Q*NO1R>@1X>T1KM153$+g*_+@F&.B3H.b +T(113$rPF[>T.B3Nk13$+

mr‹1-+TF[NO1K2*HKM1DE.B3,Z#

‘-*r$F&3G4XU+ +@(F&3G‹KRF[30Xš15>QKMX1D‰+@F[r‹1 l /`(*_+2/ 1Y/ *_3P+2+T.H,$1KTDE>@F&t1YF&3\+T(F‚K *_>@+TF‚DEh&1YFsKR*;XF&1DE1`._b

/2.J>@ˆ0+@(*+`F‚K2tU!F[h[+R.J3 +T.BX._b +@(!14I-j N*JD (!F[31 *_3,OXhs*_3K2+T.UKC14F[+QKR>@1*h +TF[NO1D*X*_t!F[h[F&+@F&1KE#

2(1H+Q*JKMˆF‚K+@.OtU!F[h‚,h[.PD*hy>T1X>T1KM13P+Q*+@F&.B3K`._b +@(!1>@.Jt.B+413Pr$F&>@.B3!NO13P+ +T(*_+“D*_3‡t1HUKC1,

+@. ,$uP3*_NOFsD*_h&h[u:N*_XHb>@11 KMX*BDE1‹lPXh‚*3*_3,\UXZ,*_+T14+@>Q*AC1DE+@.J>@F&1KE#' 1UKM1 +@(1-/ .B>g,\+@>g*_AC1DE+T.B>@u

*_3,{3.J+4F[+@F&31>g*_>@uS+@.N*>@ˆ+@(10,PFZ1>@153DE1HF[3‡+TF[NO1 K@D*h[10*3,{F&3‡Xh‚*33F&3GSDE.BNOX!h[1„PF[+Cu‹# 2(1

>@15X>@1KC13P+Q*_+TF[.J3‡/210*_>T1\GJ.BF&3G+@.S,$1KTDE>@F&tš1HFsKYh[.PD5*_h *_3,c,$.‹1K43.B+:D*>@>Tu‡*_3Pu‡KC1N*3$+TF‚DK‰#Ž+-FsK

*HXU>@1:r‹.Bh&UNO1+T>@F‚D`>@15X>@1KC13P+Q*_+TF[.J3 .bR+@(!1;b>@151KCX*BD‰1H*_>T.BU3, +@(!1>@.Jt.B+l *BK`NO1*BKCU>@1,^tPuO+T(1

KM+T1>@15.-KCuKC+@1N.r‹1>+T(1Rh‚*JKM+b1/KM1DE.B3,!KE#?Ž+FsK +T(1>@1‰b.B>@1 *DE.BN0tF&3*_+TF[.B3 ._bTlBh&1+UKK@*u‹l3.YNk.B>@1

+@(*_3}”B<O+@(>T11 ,$F[Nk13KMF&.B3*h/`F[>T1Eb>Q*NO1KEl/`(!F‚DQ(7F‚KUKM1,‡+@.^KMUXX.J>@+4+@>g*BDQˆ$F&3GO._b *3 F&+@F[3!1>Q*>@u

/`(FsD (‡(*JK;t113cKC1+4UX *BK4*OGB.‹*h2tPu‡*3.B+T(1>l KMh&._/ 1>-X>T.PDE1KTKE# 2(10>@1X>T1KM153$+g*_+@F&.B3‡FsK“+T($UK

h[.PD5*_hlPt.B+@(F&3KMX*JDE1:*_3,0F[3O+TF[NO1‹# 2(1 UKM14._b+T(F‚K2h&.PD*_hZ+@>g*_AC1DE+T.B>@uHX!h‚*_3!3F[3GHFsK +T.0*_h[h&./)+@(FsK

X>T.PDE1K@KR+@.:G‹*_+T(1>RF[3Pb.B>@N^*_+@F&.B3k*_t.BU!+2+@(!1-*JDE+@U*hKC+Q*_+T1Y.b+@(113Pr$F&>@.J3NO13P+*_3,0F[30X*_>T+@F‚D‰Uh‚*>

*_t.JU+`1r‹13P+QK?+@(*_+`D‰.BUhs,\3.B+`t1Yb.J>@1KC113lPKCUD (^*BK +T(14*XX>@. *BDQ(._b *;Nk._rPF[3!G;.JtKM+g*BDEh&1‹# 2(FsK

h[.PD5*_h?>T1X>@1KM13P+Q*+@F[.J3‡F‚K‰l .B3c.B310(*3,{X*BK@KC1,‡._r‹15>-+@.O+T(1H(F&GB(1> h&1r‹1h l KCh[./21> X>@.PDE1K@K4*_3,Zl

.B3k+@(1.B+T(1>R(*_3,\UKM1,\F[NONk1,$Fs*_+T1hnu‹lF[3k*;>T1*BDE+TF[r‹12b *JKM(F&.B3lP+@.:DE.BX1-/`F&+@(0X.‹K@KCF[th&14DE.Jh[h[FsKMF&.B3K

*_3, +@.SXš15>Mb.B>TN +T(1*JDE+@U*h2>T.Bt.B+ DE.B3P+@>T.Bh2.JX1>Q*+@F[.J3K+@(*_+h&1*B,‡+@.c*SK@*+@F‚Kbu$F&3G7X*_+@+T1>@3c._b

NO.J+@F[.J3KE#

Stereo Vision

Local Mapper Motion

Estimator

Path Planning

Supervisor

yF[GJU>@10% Y>gD (!F[+@1DE+@U>T1Y.b?.JU> KMu!KM+@15N

')(13c.BU!>->@.Jtš.J+NO.r‹1K‰lš/ 1HN\UKM+>@1GJF‚KM+T1>4KCUDDE1KTKMF&r‹1h[.PD5*_hyb>g*_NO1KX>@.rPF‚,$1,7tPu7+T(10KM+T1>@15.

KMu!KM+T1N *_3,\1KM+TF[N*+@1 +@(1NO.B+TF[.J3H._b+@(!1Y>T.Bt.B+5# 2(FsK?F‚K?X1>Mb.J>@NO1,tPuH+@(!1YNO.J+@F&.B3H1KC+@F[N^*_+@.J>l

/`(FsD (01KM+TF[N*+@1K+@(!1-NO.r‹1N013‹+yUKMF[3!G;+C/ .ˆPF[3,K?.bF[3$b.J>@N*+@F[.J3K +C/2.:DE.J3KM1DEU+@F&r‹1 1BI /`F&>@1

b>Q*NO1KR*_3, +@(!1-.P,$.BNk1+@>TF‚D,*_+g*HDE.BNOF&3G;b>T.BN +@(1:KC13KM.J>QK2.J3 +T(1-/`(151h‚KR._b +@(1->T.Bt.B+5# 2(FsK

1KC+@F[N^*_+@.J>4UKM1K4+@(1k.P,$.BNO15+@>@u‡F&3$b.B>TN*_+TF[.J3 *JKH*_3F[3F&+@Fs*_h?NO.J+@F&.B3c1KM+TF[N*+@F[.J3 +T.7N*+QDQ( +C/ .

DE.J3KM1D‰U+@F&r‹1KC+@1>T1.\b>g*_NO1K‰l *3,7+@(!13‡+@(1HN^*_+QDQ(F&3GH>@1KCUh[+gKY+@.k>@1Em3!1+@(!F‚KY1KC+@F&N*_+TF[.B3 # 2(FsK

X>T.PDE1K@K-FsK-,$1KTDE>@F&tš1H15h‚KM15/`(1>@1 …Bd‹l ?dJp_t ?*3,‡X>@.rPF‚,$1K *_+4+@(10KT*_NO1:+@F&NO1:*_3c1KM+TF[N*+@1

._b+T(1NO.B+TF[.B3k*_3,O*;NO1*BKMU!>@1`._bF&+QK?U3D‰1>@+g*_F[3P+Cu‹#Žb+@(!1-3$U!N\t1>?.bN*+QDQ(1,HKC1GBNO153$+gKyF‚KR+@.‹.

KMN^*_h[h‹.J>?3PUh[hl.B3h&u;+T(1`.P,$.JNO1+@15>F[3$b.J>@N*+@F[.J34/`F[h[hPt12UKC1,H*BK+@(1 1KM+TF[N*+@1R._b+T(12Nk._r‹15NO13‹+#

™46 < < 13MLF qBIO;2t68JZJ

WGR3u ; 0L( 8 2 9 % 91918

2(14XU>TX.‹KM1:._b +@(1 N*_XON^*_ˆPF[3G:X>@.PDE1K@K2F‚K2+T.HtUF&h‚,O*3, UXZ,*+@1:*”BI9N*_Xk1*BDQ(O+@F&NO131/

,*+Q*Yb>@.JNƒ+@(1`rPF‚KCF[.B30KC13KC.B>QKR*>@1 *rJ*F[h‚*th[1 l‹KM.4F[+qKM(.JUh‚,H.BX1>g*_+@1F[3DE>@1Nk13$+g*_h&hu # 21KCF‚,$1KElB+T(1

>@15X>@1KC13P+Q*_+TF[.J3‡UKM1, b.J>-+@(!1HN*_X‡N*ˆ$F&3GkKM(.JUh‚,7>T11DE++@(!10KM+@>TUDE+TU>@1H.b`+@(!1H13$rPF[>T.B3Nk13$+

+@(!1`>@.Bt.J+ (*BK?KC1130KM.b*_>l *_3,0KM(.JUh‚,t1KCUF[+g*_th&1 b.B> 3*r$F&G‹*_+TF[.J3HXU>@X. KM1K‰# 2(FsK Nk1*_3K+@(*_+

/21:/2.JUh‚,7h&F[ˆ‹14+@.OˆP3./9t.B+@(c+@(!1b>@11 KMX*JDE10*_3,7+T(1H.BtAC1DE+-KC(*_X1K4*3,7X.‹KMF&+@F&.B3K-F[3‡.J>Q,$15>

+@.kXš. KMF&+@F[.J3‡+@(1H>T.Bt.B+4X>T1DEFsKM1h&u\F&3‡F[+QK`13Pr$F&>@.J3NO13P+`*3,7+@.OX!h‚*_3c*kK@*b1H+T>Q*_AC1DE+@.J>@u‹# yF[3*h[h&u‹l

/214/ .BUhs,^h&F[ˆ‹1+@(*+Y+T(14/`(.Bh&14X>@.PDE1K@KDE.BUhs,Otš1 *BDQ(F[15r‹1,”F&3S*\KCN*_h&h *NO.BU!3$+`.b?+TF[NO1‹#

Ž3)+@(1b.Bh[h&./`F[3GKC1DE+TF[.B3KEl`/21SX>@.JXš. KM1 .J31cKM.Jh[U+TF[.J3)/`(F‚DQ(}UKC1K\+T(1cIY1hs*_U3*u +@>@Fs*_3

GBU!h‚*_+TF[.J3#}'c17/`F[h&h`1„PXh‚*F[3/`($u +@(1 IY1h‚*U3*u >TF‚*3GBUhs*_+TF[.B3 t*BKC1,>T1X>T1KM13P+Q*+@F&.B3}FsKH*_3

*_X!X>@.JX>@Fs*_+@1k.B31‹# Yh[h?+T(1,$FZ1>@13P+:KM+T1XK“b>T.BNo+T(1*_>T>@F&rB*h2.b * 31/ KC1+.b 1BI KC+@1>T1.7h&F[31

KM15GBNO13P+QKR+@.H+T(1-m3*_h>@1KMUh&+*_3UXZ,*+@1,r‹.Bh&UNO1+T>@F‚D”BIN*XQlP/`F[h&h t1H,$1K@DE>TF[t1, #

Y./4lBh[15+UK?KM1512(./ +T.-UXZ,*_+T1`+@(1_”BI}N*X4/`(13H*31/KC1+?._b 1BI}h[F&31 KM1GJNO13P+QK t1DE.JNO1K

*rJ*_F[hs*_th&14+@.H+@(!1;N^*_XN*_ˆPF[3!GX>T.PDE1K@K‰#

Project the 3D segments on the ground

Merge the new 2D segments with the previous ones.

Constrain the set of 2D Segments for the Delaunay Triangulation Update the Previous

Delaunay triangulation of the 2D constrained segments

Mark the empty triangles, compute the free space

Locate obstacles as convex polygons 3D Line segments from the stereo, estimate of the last movement

Map of the free space, locations of the obstacles

yF[GJU>@1\” 2(14N*XON*_ˆPF[3!GX>T.PDE1K@K*_>gD (!F[+@1DE+@U>T1

WGR H 2 (.*\%5³( WJ & ³ %J($&24%¡( ³ 2M-O%

'c10*BK@KCUNO1H+T(*_+-b.B> +@(1HNO.JNO13P+l+T(1HGB>T.BU3,‡.B3c/`(FsD (‡+T(1H>@.Jtš.J+4F‚K-NO.r$F&3G0D*3 t10DE.J3

KMFs,$1>T1,7*JK Z*_+5# 2(F‚Kh[.PD*h?Xh‚*31FsKYˆP3./`3‡+@(>T.BUGJ( D*h[F&t>Q*+@F[.J3‡*3,7UKM1,‡*BK-*KMU!XX.B>@+-b.J>

.BU!>Y>T1X>@1KM13P+Q*+@F[.J3#

' 1X>T.BAC1DE+?.J3O+@(!1YGJ>@.BU!3,O*_h&h1BI)h[F[3!1KM1GJNO13P+QKy.Jt+Q*F[31,b>@.JN KM+T1>@1.4/`(FsD (0h[F&1`t1+C/2113

+@(!1\GJ>@.BU!3, *_3, +@(1HX!h‚*_3!1\X*>Q*h[h[15h?+@.SF[+4*+H*O(1F&GB(P+;1 U*h?+@.S+@(*+4._b`+T(10>@.Bt.J+# 2(1 1BI

KM15GBNO13P+QK/`(!.‹KM1c13,PXš.JF[3P+QKO*>@1S1F[+T(1>0.B3).B>kU3,$1>k+@(1SGB>@.JU3,)Xh‚*31c*_>T173.J+X>@.BAC1DE+@1,

t1D*UKM1-+T(1u0*>@1>@115„$F&.B3K?.B> N*_>Tˆ$F&3G‹K.B3O+T(1YGJ>@.JU3,*_3,k,$.B3 †+RN*_+gD (0*3$uH>T1*_h.BtKM+Q*JDEh[1 #

' 1R(*r‹1?+T($UK >@1,$UDE1,-*2+T(>@11 ,$F&NO13KCF[.J3*_h/`F[>T1Eb>Q*NO1>@1X>T1KM153$+g*_+@F&.B3-+T.*2+C/2. ,$F&NO13KCF[.B3*_h

.B3!1‹# .0(1h&XGBUFs,$14+@(14>@1*B,$1>KRU3,$1>gKM+g*_3,$F&3Gl/21:KC(./ F&3^mGBU>T1(1H*_3SF[N*GB1-.bR*HX*>@+`._b

*O+Cu$XFsD*_h K@D‰131+T(*_+4+T(1H>@.Jtš.J+4(*BK-+@.SDE.JX1\/`F&+@(l*3,‡+@(1HX>T.BAC1DE+TF[.J37.B3c+@(!1HGB>@.JU3, .‹.B>

._bq+@(1 1BILh[F&31:KM1GJNO13P+QK2>T1DE.B3KM+@>TUDE+T1,^b>T.BN KM+T1>@1.!#2'c13.J+@F‚D‰1+@(*_++@(FsKf>T1X>@1KM13P+Q*+@F[.J37FsK

X>T.Bt*th[u3.J+`F[NONO1,$F&*_+@15h[uUKM1EbUh l$t1F&3GOKC+@F&h[h ‹UF&+@14DE.JNOXh&1„HGB1.JNO1+T>@F‚D5*_h[h&u‹#

101 Segments

yF&GBU>T11 +Cu$X!F‚D*h F[3,$.‹.B> K@DE15314*_3,^+T(14X>@.JAC1DE+@F&.B3O.bR+@(!11BIiKM1GJNO13P+QK?X>@.P,PUDE1,Ob>@.BN *

KM+T1>@15.Hr$F&1/ .b?+T(F‚K`KTDE131

bU!>@+@(!1>4F[+@15N ._bF&3$+T1>@1KM+-FsK“+T(*_+4F&3 +T(1\X!>@.PDE1KTK;.b>@1DE.J3KM+T>@UD‰+@F[3!G 1JI h[F&310KM15GBNO13P+QK‰l

/21^*_h‚KC.cDE.BNkXU+@1k*7NO1*JKMU>T1O._b +@(!1F[>4U3D‰1>@+g*_F[3P+Cu‹# 2(11„!*BD‰+NO1*JKMU>T1O,P1X13,K.J3/`(*_+

ˆPF[3, .b->T1X>T1KM13P+Q*+@F&.B3/21*>@1UKCF[3G7b.J>H+@(1Oh&F[31^KM1GJNO13P+QK …Bd$l dBpB* `tU+HF&+H*_h&/`*u!K

>@15X>@1KC13P+QK`+@(10*NO.BU3P+._bD‰.B3$mZ,P13DE1:/21H(*r‹1F&3‡*KMX1D‰FmZD:KM1GJNO13P+# 2(!F‚K-DE.J3$mZ,$153DE1HF‚K*

bU3D‰+@F[.J3 ._b+T(10>@1h&F‚*_t!F[h[F&+Cu .b+@(1krB*>@F&.BUK1h&1NO13P+QK`F[3c+@(!1OX>@.PD‰1K@KCF[3G D (*F[3 KMUDQ(*JKEl?1,$GJ1

,$1+T1DE+TF[.B3 lXš.Jh[uPGB.J3*_hR*XX>T._„PF[N^*_+@F&.B3l!D*_h&F[t>g*_+TF[.B3S._bR+T(1+T>@F&3.PDEUhs*_>KM+T1>@15.\>TF[GlZ*BK`/215h[h*BK`._b

+@(!12GB15.BNO1+T>@u._b+T(1`KTDE131 KM1GJNO13P+QK /`(FsD (:*_>T1 b *>?*>@12h&1K@K>@15h[F‚*th[1 +@(*30KM1GJNO13P+QKš/`(!F‚DQ(H*_>T1

DEh&.‹KM1 Q#?Ž+?FsK?>@1X>T1KM153$+T1,;tPuH*:KMuPNONO1+T>MFsDy/215F[GB(P+?N*+@>TF[„ZlB*h‚KM. D*_h&h[1,:KM.JNO1+@F&NO1K*4DE.rJ*_>TF‚*_3DE1

N*+@>@F&„-Fbš*“X!>@.Bt*_tF&h[F‚KC+@FsD F[3P+@15>@X>T1+Q*+@F[.J34F‚Ky>T1 U!F[>@1, l_/`(!.‹KM1KCF1 F‚Ky+T(*_+ ._bZ+@(1`>T1X>T1KM13P+Q*+@F&.B3l

*_3,/`(.‹KC1,$F‚*GB.B3*_h1h[15NO13P+QK *>@1 tF[G-Fnb+@(!1`DE.J>@>@1KMX.B3,PF[3G4X*>Q*_Nk1+@1>gK F&3+T(12>T1X>@1KM13P+Q*+@F[.J3

F‚K2U!3DE1>T+Q*_F&3l*3,7KCN*_h&h .J+@(1>T/`F‚KC1‹#

>@.BN +@(!1,1BI >@15X>@1KC13P+Q*_+TF[.J3 .b-+T(1Oh[F&31kKM1GJNO13P+QK4*_3,c+@(1D‰.B>@>T1KMX.J3,$F[3!G NO1*BKMU!>@1H._b

U3D‰1>@+g*_F[3P+CuHF[+ F‚K X. K@KMF&th[14+T.HDE.JNOXU+T1-*>T1X>@1KM13P+Q*+@F[.J3\b.B> +@(14X>T.BAC1DE+T1,0KM1GJNO13P+QKR*_3,O+T(1

DE.J>@>@1KMX.B3,PF[3G0U3DE15>@+Q*F[3P+Cu KM11:KM1DE+@F&.B3 1$#1$#E” Q#RWY3DE1>@+g*_F&3$+TF[1KR*_>T1HDE.J3$r‹153F[13P+@h&u>T1X>@1KM13P+@1,

tPu‡1h[h&F[XKC1K-F[3c+@(10Xh‚*31O/`(15>@10+@(1ucD5*_3 t1O+T(.BUGJ($+H.b *BK4GB1.JNO1+@>TF‚DUX!Xš15>t.BU3,KE# .J>

1„!*_NOX!h[1‹lmGBU!>@1- KM(./K2+@(!1Y1h&h[F&XKM1K?>T1X>@1KM13P+@F&3G-+@(!1-U3DE15>@+Q*F[3P+@F&1Ky._b+T(1-NOFs,$X.BF&3$+gKy._b+T(1

X>T.BAC1DE+T1,7KC1GBNO153$+gK ._bRmGJU>@11P#Y'c13.J+@1:+@(*+4KM1GJNO13P+QK2/`(FsD ( *_>T1;b *_>b>@.BNv+T(1>T.Bt.B+:*_>T1

NO.J>@1-U3DE1>@+g*_F&3 +T(*_3S+@(. KM14/`(FsD (S*_>T1:DEh[. KM1‹#

‘-*r$F&3GKCUD (c*ONO1*BKMU!>@1FsK-F[NOX.B>T+Q*3$+t1D*_UKM10F[+4*h[h[./KYX!>@.PDE1KTKM1K-+T(*_+4.BX1>g*_+T1\.J3 +T(1

>@15X>@1KC13P+Q*_+TF[.J3K?+@.4GBF&r‹1`NO.B>T1`/21F&GB(P+?+@.:>@1h&F‚*_t!h[12Nk1*BKCU>@1Nk13$+gK +@(*3O+@.4.J+@(1>gKE# Y301„!*_NOX!h[1

._b KCUD ( *YX>T.PDE1K@KF‚Ky.J312+T(*_+?1KM+@F&N*_+T1K +@(!1`>@.Bt.J+?1GB.JNO.B+TF[.J3 ?…BdPl …B…$l ?dJpB* !*_3,+T(1

.BtKM+Q*JDEh[1KyNO.B+TF[.J3 dBpBDJl ?dJp_t #?Ž3H+@(!F‚K*_>@+TF‚DEh&12/ 12UKC1`+@(FsKyF[3$b.J>@N*+@F&.B3+T.-(1h&XX>T.P,$UDEF&3G

*KMF[NkXh[Fnm1,`r‹1>QKCF[.B34.b+@(1~”JIN*X;.b$mGBU>T1 1$# 2(12N^*_F[3 >@1*JKM.B34b.B>X>@.P,PUDEF[3!G-KCUD (:*`N*_X4FsK

+@(*_+._b DE.B3DEF‚KC131KTKE#?•‹F[3DE1R.BU>?GJ.‹*h$FsK +T.-X>@.P,PUDE1`*`N*X._b!b>@11 KMX*BDE1‹lJ/21 *_>T12U3F&3$+T1>@1KM+@1,“F&3

>@1,$U3,*3$+ F[3$b.J>@N*+@F[.J3HKCUD ( *BK?+@.‹.4N*3$u lGB1.JNO1+@>TF‚D*h[h&u-KCF[NOF&h‚*_>5lKM1GJNO13P+QKF[3:+@(1KT*_NO1 *_>T1*

+@(*_+X. K@KMF&th[u *_>TF‚KC1”b>T.BN * (!F[GB(!h[u‡+@15„$+TU>@1,‡X*+@+@15>@3#7•‹U!XX>@1K@KMF&3G7>T1,$U3,*_3P+F&3$b.B>TN*_+TF[.B3

/`F[h&h *_h[h&./ƒUK+@.Sˆ‹11X .J3h[uc+@(1kF[3$b.J>@N*+@F[.J3 >T1*_h&h[u‡>T1h[1rJ*3$+4+@.‡+T(1O+g*BKMˆ *_+H(*_3,*_3,/`F&h[h

b *BD‰F[h[F&+Q*+@14*3,7KCXš151,OUX bU>T+@(15>`X>@.PDE1K@KM1KE#

WGR W Y 01(+*\%'³( H 0€Yg*,( *\%J( ³ 0L(.*\%5³( % & ( 27APY 2€

91:

(

918

& 9 ³ C 9

%J($&

`+21*JD (O3!1/’DEu!DEh[1‹l‹+T(1M”BI9KC1GBNO153$+gK D‰.BNOF&3G-b>@.BN +@(1-hs*BKC+`/`F[>@1‰b>Q*_Nk1*_>@14NO15>@GB1,”/`F&+@(O+T(1

KM15+Y.b~”BIƒKC1GBNO153$+gK2/`(F‚DQ(>@1X>T1KM153$+`/`(*+-+@(!1>@.Jt.B+-(*JK-KM15137F&37+@(!1X>T1r$F&.BUK`D‰uDEh&1KE# 2(1

N*XO._bR+T(1\”BIiKM1GJNO13P+QK FsK2UXZ,*_+T1, #

21Eb.B>T1:KM+@U,$u$F&3GH+@(!1;Nk1>@GJF[3G:X>@.PDE1K@KEl/ 1:,$1K@D‰>@F[t14(./’h[F[3!1KC1GBNk13$+gKR*_>@1:>@1X!>@1KC13$+T1, #

[

$

}~|

"

š

-”BI9.B>1JI `h[F&314KC1GBNO153$+`FsKfUKCU*_h&h[u>@15X>@1KC13P+@1,\tPuF[+gKf13,$X.BF[3P+QK-*3,-*3,7+@(!1F[>

DE.rJ*_>TF‚*_3DE1N*_+T>@F‚D‰1K H*_3,#}• F[3DE1k+@(113,PXš.JF[3P+QK.b4*cKM1GJNO13P+:*_>@13.J+\>T1h[Fs*_th&1‹ly/ 1

D*33.B+ ,$F[>T1DE+Th[u4UKM1 +@(1NiF[30NO.‹KC+ D*BKC1KE#2€R.J3KMFs,$1>@F&3G4+@(1h[.J3GBF&+@U,PF[3*h*_h&.B3G:+@(1 KM1GJNO13P+

X.‹KCF[+@F&.B3H.bZ*Yh[F&31 KM1GJNO13P+ FsK N0UDQ(-h[1KTK X!>@1DEFsKM1R+@(*_3H+@(!1 +T>Q*3KMr‹1>gKM1 .B31‹lJ.B31 N*u-UKM1 F[3KC+@1*J,

F[+gK4KMUXX.J>@+@F&3G7h&F[31 #\‘._/ 1r‹1>5l*b+T1>4+@(FsK;*tKM+T>Q*JDE+@F&.B3 / 1\h&.‹KM1kDE.JNOXh[15+@1h&uO+@(10h[.J3GBF&+@U,PF[3*h

F[3Pb.B>@N^*_+@F&.B3l?/`(!F‚DQ( F‚KUKC1EbUh`F&3 N*_3Pu D*BKC1KHKMUDQ( *BKHN*_+gD (!F[3GS*_3,bUKCF[.B3 # U!>@+@(!1>l?+T(1

U3D‰1>@+g*_F[3P+Cu-._bZ+T(1KMUX!Xš.J>@+TF[3G4h[F&31`,P.‹1Ky3.J+ >T11D‰++@(*+ .bZ+@(1 KM1GJNO13P+lnlB+@(!1KMUXX.J>@+@F&3G

h[F&31-.b *”r‹15>@uOU3DE1>@+g*_F&37KC1GBNk13$+ N*u\t14h[1K@K`U3DE1>T+Q*F[3+@(*37+T(1:KMUXX.J>@+@F&3GOh&F[31-.b *Hh[1KTK

U3D‰1>@+g*_F[3^KM1GJNO13P+# .B>2+T(.‹KC1->@1*BKM.J3KEl/ 1-(*r‹1X>@.JXš. KM1,S*;315/)>@1X>T1KM153$+g*_+@F&.B3\b.J> 1JIah&F[31

KM15GBNO13P+QKR/`(F‚DQ(OFsK`*”+T>Q*B,P1 . t1+C/ 113Oh&F[31K *_3,SKM1GJNO13P+QK (*JdBpl dBpB* #

Ž3}.BU!>O>@1X!>@1KC13$+g*_+TF[.B3 l2+@(1 KMX(!1>@FsD*_hDE.‹.J>Q,$F&3*_+T1K *_3,c*_>T1‡UKM1,+@. >@1X>T1KM153$+H+T(1

,$F&>@1DE+TF[.J3 .b * h[F&31kKM1GJNO13P+l*_3, +T(1\NkF‚,$X.BF&3$+

+@.‡h&.PD*_+T1\+T(1KM15GBNO13P+# 2(!1Oh[13!GB+@(c._b

+@(!14KM1GJNO13P+RF‚K2,P13.B+T1,\tPu #RŽb/214,P13.B+T1

t$u

z

!!" $# +T(1-3.J3 3.B>TN*_h&F1,

,$F&>@1DE+TF[.J3Or‹1DE+T.B>gl$+T(14U3F[+,PF[>@1DE+@F&.B3Or‹1D‰+@.B>`tPu

)z&%'Qz('

l )* +#ytPu-,-l/ 1;GJ1+

./

0/1

*_>gDDE. K 2

3

254769874

Fnb:<; p

”>= *_>gDDE. K 2

3

254769874

.J+@(1>T/`F‚KC1

*_>gDDE.‹K ?

3

254769874@6 ? 4 8

T%

2(1:DE.rJ*_>TF‚*_3DE14N*_+T>@F[„A

,

.bB, F‚K2GJF[r‹13 l +T.\+T(1-m>QKC+Y.J>Q,$15>*_XX>T.„$F&N*_+TF[.B3 l$tPu

,

DC

,

C z

&E

C ,

C z # -”

/`(1>T1GF

,

F E

F‚KR+@(1IH‹*BD‰.BtFs*_3ON*+@>TF[„.bJ,}/`F[+T(O>@1KCX1DE+R+@.

z

l$*3,-&E

LKME#2• F[NOF&h‚*_>Th[u‹l/ 1

D*37D‰.BNOXU!+@1-+T(1:DE.rB*>@Fs*_3DE14N*+@>TF[„<&N.bR+@(14U3!F[+,$F&>@1DE+TF[.J3Or‹1DE+T.B>

#

2(14NO.‹KC+2F&NOX.B>@+g*_3P+2X*>@+._b .BU>`>T1X>T1KM13P+Q*+@F&.B3F‚KR+@(14NO.P,$15h[F*_+@F&.B30._bR+T(1-U3DE1>@+g*_F&3$+Cu

._bq+@(14NOFs,$X.BF[3P+# 2(14NOFs,$X.BF&3$+ ._b *h&F[314KC1GBNk13$+ F‚K2NO.P,P1h[1,k*BK

O P

KQ %

”(KSR

1

/`(1>T1GR FsK;*O>g*_3,$.JN rJ*>@F‚*th[1 #OŽ37b *BDE+5ly1 ‹U*+@F[.J3 1KT*u!K;+T(*_+4+T(1\NkF‚,$X.BF&3$+ ._b*KC1GBNO153$+

N*u0rB*>@uO>g*_3,P.BNOh&uO*>@.BU!3, F&+QK2X.‹KCF[+@F&.B3 PTKS %

”\*h[.B3!G\+T(1:,$F[>T1DE+@F&.B3O.bR+@(!1KC1GBNk13$+5#

FsKKM.JNO10Xš. KMF&+@F[r‹1^K@D*h‚*_>5#cŽ 3 .BU>HF&NOXh[15N\13P+Q*+@F&.B3l

F‚K;>T1h‚*+@1,c+@.‡+@(!1Oh[13!GB+@( 2.b-+T(1

KM15GBNO13P+l

/21DQ(.‹.‹KC1

pP8E” g# 2(!F‚KHK@*u!K+@(*+O*7h&.B3G KM15GBNO13P+4F‚KN0UDQ( h&F[ˆ‹1h&u

+@.St1\t>T.Bˆ‹13cF[3P+@.^KMN*h[h[15>-.B31K“F&3 .J+@(15>;rPF[15/KE# 2(!1ODE.rJ*_>TF‚*3DE1HN*+@>@F&„7.b

F‚K-GBF&r‹13‡t$u

KC11 ?dJpB*Pl(*BdJp

TKS %  K

&N K -

#

g8 

2(1HU3DE1>@+g*_F&3$+CuF[3‡+T(1h&13GB+T(‡._bf*OKM1GJNO13P+YFsKY3.J+-NO.P,$1h&1, t1D5*_UKC1HF[+-FsKY3.B+ >@1 ‹UF[>T1, F&3

.BU!>-*h[GB.J>@F&+@(N‡#

{”BIƒh[F&31HKC1GBNO153$+ D*_3ct1Hr$F&1/21,7*BK-+T(1H.B>T+@(.JGB.B3*_h2X!>@.BAC1D‰+@F[.J3‡._b *,1BIƒh[F&31HKC1GBNO153$+

.B3 *”X!h‚*_3!1‹# …” 1HN*+@>@F&„ D*_3Stš1:UKM1, +@.k,$1KTDE>@F&t1;+T(F‚K`X>T.BAC1DE+TF[.B3 lnl/21 D*_3S>@1hs*_+@1

*(1BI9X.BF[3P+

/`F&+@(OF&+QK2X>T.BAC1DE+TF[.B3

(15>@1:KMUtK@DE>TF[X+gK`*_>@14UKC1,\+T.O,$153.B+T1-+@(1:,$F&NO13KCF[.J3l

/`(FsD (/`F[h&h t14.BNOF&+@+T1,\Fb3!.O*N\tF&GBUF&+Cu tPuO+@(1-b.Jh[h[./`F&3GH1 ‹U*_+TF[.B3

8 <

.B>`1„!*_NkXh[1‹lJFb/21 /`*_3P+`+@.HX!>@.BAC1D‰+`*(1BIX.BF&3$+`.J3+@(1-X!h‚*_3!1

p$l+T(13

% p p

p p %

#

'c1Y>T1X>@1KM13P+B”BI’h[F[3!1KM1GJNO13P+QKKMF[NkF[h‚*>@h&u;+T.*1BI h&F[31KC1GBNO153$+gKElB+T(*_+RF‚K‰lP*@”BI)h[F&31 KM1GJNO13P+yFsK

,$1KTDE>@F&t1,-t$u-F&+QK*3GBh&1`/`F[+T(4+@(1 *_„PF‚KlF[+gK NOFs,$X.BF&3$+

l*_3,4F[+gK h[13GJ+@( # 2(. KM12X*_>Q*NO1+T1>QK

D*3 t1:1*BKCF[h[ukDE.JNOXU+T1,\t*JKM1,.B371 U*+@F&.B3S<$#R' 1:DE.JNOXU+T14*_hsKM.0+@(14rJ*_>TF‚*3DE1

b.B>IH*_3,

+@(!1HDE.rJ*_>TF‚*_3DE14N*_+T>@F[„ Lb.B>

# .B>1„!*_NOXh&1‹l

4

# # 2(1:U3DE1>T+Q*F[3P+Muk.B3FsK

3.J+YNO.P,P1h[1,Z#

[‚

""{

"

š

-

|

‹

~

`+Y+T(F‚K`X.BF[3P+lZ/21HKCUXX.‹KC1\+T(*_+ /21(*r‹1 *3”BILN^*_X7>T1X>T1KM13P+@F&3GH/`(*+;+T(1>T.Bt.B+4(*JK-KM1513

X>T1r$F&.BUKCh[u‹lJ*_3,\*-KC1+?.bs”JI)KM1GJNO13P+QK>@1X!>@1KC13$+TF[3G+@(!12h‚*JKM+?F&3$b.B>TN*_+TF[.J3K +T(1`>@.Jt.B+R(*BKRKC113#

W-KMF&3G4+@(11KC+@F[N^*_+@F&.B3:._b+@(!1`NO.r‹1NO1‰3$+GJF[r‹13HtPuH+@(1`Nk.B+@F&.B301KM+TF[N*+@.J>l+@(1X>@15r$F&.BUKRKM15+

._b KC1GBNk13$+gK?._b+@(!1-.Bhs,\N*_XkD*3Ot1-+@>g*_3Kb.B>@Nk1,”F&3O+@(!14DEU>T>@13P+ DE.‹.B>g,$F[3*+@14KCuKC+@1N l‹*_3,O+T(1

NO1>TGBF&3GX>T.PDE1KTKYD*3tš1 *_+T+@1NOX!+@1,H/`F[+T(7+@(!1KC1+`._b 31/`h&uONO1*JKMU>T1,OKC1GBNO153$+gKE#

2(12mGJU>@1 14KM(!._/K DEh[1*>@h&u;+T(*_+ +@(1`X>T.BAC1DE+T1,KC1GBNk13$+gKDE.B3P+Q*F[3H>T1,$U3,!*_3P+?F[3$b.J>@N*+@F&.B3

KM.JNO1`D‰.Bh[F&31*>RKM1GJNO13P+QK*_>T1`r‹1>@u:DEh&.‹KM1 l‹KM.JNO1KM1GJNO13P+QKKM115N]+T.4>@1X>T1KM153$++@(1KT*_NO1`X!($u!KMFsD*_h

.BtAC1DE+`tU+`1„PXh&.P,$1,OF[3SKM15r‹1>Q*h X*>@+QK‰lšF&3KM.JNO1:*_>T1*BK`+@(1:KC1GBNO153$+gK *>@1:r‹1>@uk,$13KC1‹#[#&#

Ž3H+@(12b.Jh[h[./`F&3Gl‹/21 ,$1KTDE>@F&tš1`(!._/ +@.4F[3P+@15h[h[F&GB13P+@h&uYNk1>@GJ1 GJ1.BNk1+@>TF‚D*h[h[u“KMF[NkF[h‚*>KM1GJNO13P+QK

F[3+T(1:K@*_Nk14*_>T1*$l/`(FsD (S*>@1‹l$b.J>`1„*NOXh&1‹l‹X>@.BAC1DE+@F&.B3K2.bRKC.BNO14+@15„$+TU>@1KR.B3 */`*_h&h#

2(1SKMF&NOXh[FnmZD*+@F[.J3 FsKX1>Mb.J>@NO1,c*BKb.Bh&h[./KE# Yh[h KM1GJNO13P+QK4*_>T1SKM.B>T+@1, tPu*7tUDQˆ‹1+TF[3G

+@1D (3!F‹U12/`(FsD (0*h[h&._/K?+T.1*JKMF[h&uH*JDDE1KTKy+@(131F&GB(Pt.B>QKR._b *;KC1GBNk13$+5# 2(1-KC1GBNk13$+gKy*>@1 *_hsKM.

KM.J>@+T1,‡*BDD‰.B>Q,PF[3G+@.k+@(1F&>-.B>TF[13P+Q*+@F&.B3# 3h&u‡*”b1/iDE.JNOXU+g*_+TF[.B3K-*_>T1H>@1 ‹UF[15>@1,+@.OUXZ,*+@1

+@(!1;h&F‚KC+`._b 31F[GJ($t.J>QK`/`(13+@(14315/1”JILKM15GBNO13P+QK *_>T1H*B,!,$1,O+@.0+@(14.Jh‚,O.B31KE#

Y./ b.J>7* KM1GJNO13P+\3!.B+7u‹15+OX>@.PD‰1K@KC1,! lY/ 1 ˆP3./ +T(1 h&F‚KC+O._bkKC1GBNO153$+gK”/`(FsD ( *_>T1

31F&GB(Pt.B>QK4+@.SF[+:*_3, *_3.J+@(1>:h[FsKM+4._b KC1GBNO153$+gKY/`(F‚DQ(c(*r‹10KMF[NkF[h‚*>-.B>TF[13P+Q*+@F&.B3K-+@.SF[+5# 2(1

F[3P+@15>QKM1DE+@F&.B3._bR+T(1+C/ .\h&F‚KM+gKY*>@1:+@(14m>QKC+-D*3,$Fs,*_+T1KE# Ž bq*OKC1GBNO153$+" $# *_NO.J3G0+@(. KM10D*3,$F

,*+@1KRF‚KRKMF&NOF[hs*_>?13!.BUGJ(O+@.% 2lP/21+@(153\NO15>@GB1 *3,&

#

uPF[15h‚,$F&3G4*;3!1/’KM1GJNO13P+(' #?' 1+@(!13

t1\U!X ,!*_+@1, # 2(1k*_t.r‹10X>@.PDE1,$U>@10F‚K4*XXh[F&1,7+@.S1r‹1>Tu7D5*_3,$Fs,*_+T1O*3, 1r‹15>@uU3X>T.PDE1KTKM1,

KM15GBNO13P+#

2(14NO1>TGBF[3!G;+T1DQ(3F ‹U1`b.B>”JIah&F[31“KM1GJNO13P+QK?F‚KR+@(1:*B,!*_X+g*_+@F&.B3S._b +@(*+2b.B> 1JI]h&F[31 KM1G

NO13P+QK dBpB*l (*JdBp # 1+B$l

*_3,-tš1+@(!1YX*>Q*NO1+@15>QKy.b *KM15GBNO13P+ ‡/`F[+T(\U3DE1>@+g*_F&3$+Cu

NO1*JKMU>T1NO13P+QK

*_3,- l*_3,- #l # *_3,# +T(*_+`.bR*3.B+T(1> KM1GJNO13P+ # /`F&+@(OU3DE1>@+g*_F&3$+Cu

NO1*JKMU>T1NO13P+QK

*_3,A

# 2(. KM1-+C/2. KM1GJNO13P+QK?D5*_3Ot1NO1>@GJ1,HFb*3,\.J3h[uHFnb+@(1u0KT*_+TF‚K bu

+@(!1“b.Jh[h&._/`F&3GH>@15h‚*_+TF[.J3K2t*BKC1, .J37+T(1:j‡*_(*h‚*_3!.BtFsKY,$FsKM+g*_3DE1

P

# % K

–

# #

P KQ

#

A CV

/`(1>T1

*3,

o*>@1O+T(>@1KC(.Bhs,KE# .‹.BˆPF[3!G UX +@(1 H,PF‚KM+T>@F&tU+@F&.B3c+Q*th[1‹l/21kD*_3D (.‹. KM1

1$8…_`b.B> *2X>@.Jt*_tF&h[F&+Mu ._bdB<’/`F&+@(0%R,$1GJ>@11R._b!b>@11,P.BN *_3,

<P8dJd b.J>*fX>T.Bt*tF[h&F[+Cu

._b dB<L/`F&+@(q”,$1GJ>@11Ky._bb>T11,$.JN‡# 2(1-*t._r‹1 DE.J3,$F[+TF[.J3KRK@*uH+@(*+2/ 1YNO15>@GB1 .B3h&u0KM1GJNO13P+QK

/`(FsD (S*_>T14+@(1:K@*NO1F[3+@(14X>T.Bt*tF[h&F‚KC+@F‚D KM13KC1 Q#

`*BKC1, .J39* NOF[3!F[NHUN rJ*>@F‚*3DE11KC+@F[N^*_+@.J>l`/21cGB15+ +T(1‡b.Bh[h&./`F[3GX*>Q*_Nk1+@1>gK”b.B>+T(1

NO1>TGB1,kKM1GJNO13P+' '

K

# %

K

5 …

% K d

'

KS

KQ P KS

# @%5p

P KQ

8 @%J%

2(1u:GBF&r‹1 +T(1`.B>TF[13P+Q*+@F&.B3H*3,H+@(1 Xš. KMF&+@F[.J3\.bZ+@(1 NO1>@GJ1,KC1GBNk13$+5#y•‹F&3DE12+T(1`+C/2. KM1GJNO13P+QK

*_>T1-D‰.B3KCF‚,$1>T1,H*BK?+C/2. F[3KC+Q*_3DE1K?._b *4KMF&3GBh&1KM1GJNO13P+l+@(1U3F&.B3H._b +@(15NƒKM(!.BUhs,”t1F[+QKtš15+@+@15>

1KC+@F[N^*_+@1 # 2u‡X>T.BAC1DE+TF[3Gk+@(1013,$X.JF[3P+QK-._b +@(10+C/2.SKM1GJNO13P+QK`.B3 +T(1HNO1>TGB1,‡KC1GBNk13$+5lš/ 1

DQ(.‹.‹KC1O+@(1Hb*_>T+@(1KM+X!>@.BAC1D‰+@F[.J3K4*BK4+@(1O153,$X.BF&3$+gK;._b +@(1kNO1>@GJ1,‡KM15GBNO13P+4*3, +@(!1\h&13GB+T(

D*3c*_h‚KC.tš10D‰.BNOXU!+@1, # 2(!1 $NkF‚,$X.BF&3$+I ._b +@(1HNk1>@GJ1,KM1GJNO13P+-D*3‡t10,$1+@15>@NOF&31,

*_3,7D*37*h[/ *u!K2t11„PX>T1K@KC1,O*BK

'

K

'

@%y”

/`(1>T1FsK`*OK@D*h‚*>-*3,

'

F‚K2+@(!1U3F&+-,PF[>@1DE+@F&.B3r‹1DE+T.B>`._bq+@(1:NO1>@GJ1,OKC1GBNO153$+5# 2(1 *_t.r‹1

1 ‹U*_+TF[.J3cD*_3ct1HF[3P+@1>TX>@1+T1,7*JK-+@(10*J,,$F[+TF[.J3‡._b* ”3.JF‚KC1H.B3

'

l +@(PUKY+T(10DE.rJ*_>TF‚*_3DE1

N*+@>@F&„\.b: F‚K2GJF[r‹13OtPu

K

P

N K ' ' #

g8 @% 1

.B>`NO.J>@14,P1+Q*F[h‚K‰l+@(14>T1*B,$15>`F‚K2>@1‰b1>@>T1,\+@. ?dBpJ*l(*BdJp #

Y3)1„!*_NOX!h[1._b:>@1*_h ,*_+g* bUKMF[.J3)F‚KOKC(./`3 F&3)+@(1mGBU!>@1K< *3, –P#9IYU>TF[3G+@(!1 m>gKM+

F[+T1>Q*+@F&.B3lB+T(1KM15GBNO13P+QK._bmGJU>@1 1-*>@12bUKC1,”/`F&+@(H+T(1NKC1h[r‹1K‰# 2(!1`>@1KCUh[+gK?D*_30t1`1„!*_NOF&31,

F[3‡mGJU>@1k<$#O'c1\3!.B+@FsDE10+@(*+;+T(103$UN0t1>-.bKM15GBNO13P+QKF‚K4DE.B3KMF‚,P1>Q*th[u‡>T1,$UDE1, # 2(153 +T(1

>@.Jt.B+4NO.r‹1K*_3,7GJ1+:*\31/iKM15+-._b 1BI ,*_+g*$l/`(F‚DQ(‡*>@1HUKC1,‡+@.O1KC+@F&N*_+T1F[+gKY,PF‚KMX!h‚*BD‰1NO13P+

KC11KM1DE+@F&.B3q” Q# 2(!1`X>@1rPF[.JUKRKM1+R.bV”BI’KM15GBNO13P+QK._b+T(12mGJU>@1 <-F‚KybUKM1,”/`F&+@(0+@(1`315/),*+Q*

*b+T1>X>@.BAC1DE+@F&.B37.J3 +T(1GJ>@.JU3,‡Xh‚*31‹# 2(!1;mGJU>@10–OKC(./KY+@(!1H>@1KCUh[+gKf.b2+T(F‚K2bUKCF[.B3 # 2(1

31/’UXZ,*+@1,N*_XF‚K2+T.Ht1HDE.JNOX*_>T1,\/`F&+@(+@(14X>T1rPF[.BUK2N*_XO.b?mGJU>@1:<$#

44 Segments

yF&GBU>T1< yF[>gKM+KC+@1X RKCF[NOX!h[F mšD*_+TF[.B30._bq+@(1:KM15+`._b KC1GBNO153$+gKq.b?mGJU>@1*1

160 Segments 55 Segments

F[GBU!>@1:– RWYXZ,*+@F&3G\+T(1\”BI9N*_X# 21Eb.B>T1:*_3,*b+T1>+@(1-bUKCF[.B3

z z

')(13+T(1KM15GBNO13P+QK4*>@1NO1>TGB1, lr‹1>Tu .b+@13 +M/ .‡.B>HNO.J>@1kKM1GJNO13P+QK4*_>T1ObUKM1, F&3$+T.‡*7tF&G

.B3!1‹l 1KCXš1DEF‚*h[h[ukFbR+T(1uS*_>T1X*>@+QK._b *OK@*NO1X!($u!KMFsD*_hKC1GBNO153$+KC1137X*_>@+TF‚*h[h[uOb>T.BNo,PFZ1>@153$+

rPF[1/`X.BF&3$+gKE#ŽN*GBF[3!1+@(*_+.JU>>T.Bt.B+H+TU>@3K;F[+gK-(1*B, DEh[.PDQˆ$/`FsKM10*3, h[.‹.JˆK:*_+ *_3 1,PGB1O.b*

/`*h[h # 2(141,$GB1:/`F[h&h t1>T1tUF&h[+2tPuO+T(14NO1>@GJF[3G:X>@.PD‰1K@K*_3,Fb 1r‹1>Tu$+T(F[3G:GB.‹1K2m31 lKM(.JUh‚,

t1DE.JNO1-.B3!1;h&.B3G0KC1GBNk13$+5# 2(14X>T.Bth&1N F‚K2+@(*_+`+@(!1;h&.B3G0>@1KMUh&+@F[3!GHKM15GBNO13P+R(*BK`t113SKM1513

F[3>T1*_h&F[+Cu”b>T.BN ,$F 15>@13P+`r$F&1/`X.BF&3$+gKEl,$F 15>@13P+2h&.PD*_+TF[.B3KYF[3SKMX*BDE1‹#R'c1311,^+T.\ˆP3./’/`(F‚DQ(

X*>@+(*BKYt11537KC113b>@.BN /`(F‚DQ(>@.Bt.J+-X.‹KMF&+@F&.B3# 2(!F‚K`F[3$b.J>@N*+@F[.J3F‚K2>@1 UF&1>@1,\+@.kDE.BNkXU+@1

DE.J>@>@1DE+@h&u\+T(1-b>@114KCX*BD‰1‹l*BK`+@.0t1,P1K@DE>TF[t1,OF[3SKC1DE+@F&.B3#†#

S1

S2

S

a

b

c

Seen from 1 Seen from 2 Seen from 1 and 3

Part a seen by camera position 2 Part b seen by camera position 1 and 2 Part c seen by camera position 1

Two instances of the same segment seen from two different camera positions.

yF&GBU>T1-V ?')(13HNk1>@GJF[3G-+C/ .4KM1GJNO13P+QK‰lB+@(1`rPF‚KCF[tF&h[F&+Mu ._b1*BDQ(HX*_>T+2._b +@(1`>T1KMU!h[+@F&3G4KC1GBNO153$+

F‚K`D‰.BNOXU!+@1,

OPMqJ?|B 9q2k13MF < ?POP7»9 6 BI6qBI68J|6–M^<Hk1AOPM O k4L6 BI6H6

J q<E2t6

K

R3u !24%5&( = - : (+*\%'³s0 9 6307-O%'0 ´@( = *30^%'³4-6301(+*324%@27A0

H C 0µY

2(1 F‚,$1*2FsK +T.-D‰.BNOXU!+@1‹l*JK?*_3:F[3P+@1>TNO1,$Fs*_+@1>@15X>@1KC13P+Q*_+TF[.J3l*Y+@>TF‚*3GBUhs*_+TF[.B3:._b+T(12153,$X.BF&3$+gK

._b +@(!1M”BI9KM15GBNO13P+QKR+@(*+`(*BK2+T(14DQ(*_>g*BDE+T1>@FsKM+TF‚D`._bqDE.B3P+Q*F[3F&3GH+@(1“KM1GJNO13P+QKR*BK21,PGB1K2.b?+T(1

+@>TF‚*3GBUhs*_+TF[.B3 # 9KM1+`.bRXh‚*3*_>X.BF&3$+gKfFsKYKT*_F‚,+@.Ht1+T>@Fs*_3GJUh‚*+@1,Fb F[+QK2X.JF[3P+QK*_>@1AC.BF&31,Ot$u

3.J3F[3P+@1>gKM1D‰+@F[3!GOKC+@>Q*F[GJ($+h[F&31HKC1GBNO153$+gK`KM.O+T(*_+-15r‹1>@uO>T1GBF&.B3SF[3P+@1>T3*_h+T.O+@(1 DE.B3Pr‹1„(PUh[hFsK

*H+@>@Fs*_3GJh[1‹#

IY1hs*_U3*uk.B31:tš1D*_UKC14._bR+C/2.H>T1*BKC.B3K

Ž+`(*JKYKC.BNO1-3!F‚DE1-X!>@.BX1>T+@F&1K2>@1hs*_+T1,O+@.kKM(*Xš1 l*_3,

Ž+D*3 t1:F[NOXh&1N01E3P+@1,1 DEF&13P+@h[u #

2.B+T(>@1*JKM.B3K-*_>T1;/ 1h[h ,$1KTDE>@F&tš1,\F[3 j `dBp #

2(1SKM(*X1 X!>@.BX1>T+CuD*_3 tš1SKCUNON*>@F51,‡*BK;b.Jh[h[./K4F[3 +C/2.c,PF[NO13KMF[.J3K-tU+HF&+F‚K4+@>TU1

*_hsKM.0F[3O+T(>@151,PF[NO13KMF[.J3K KMUX!Xš. KM1:/214NO1*JKMU>T1-*H3$U!N\t1>R._bqXš.JF[3P+QK2.J37+@(!1;t.JU3,*>@u._bq*_3

.BtAC1DE+R*_3,O*BKTKMUNO1`+T(*_+ +@(FsKyt.BU3,*_>TuOF‚KyX!F[1DE15/`F‚KM1KCNO.‹.B+T(# 2(!13l‹U3,$15>2KC.BNO12b*F[>Th[u/ 1*_ˆ

*BKTKMUNkX+@F&.B3K‰l +T(1IY1hs*_U3*uH+T>@Fs*_3GJUh‚*+@F[.J3\.b+T(1-KC1+R._b NO1*BKCU>@1,;X.BF&3$+gK2DE.J3$+g*_F[3K *-X.Bh&u$GJ.B3

+@(*_+`*XX>T._„PF[N^*_+@1Kq+T(1tš.JU3,*>@uKC(*_X14/215h[h#Žb+T(*_+`X.Jh[uPGB.B3^D*_3Ot1-1*BKMF&h[uH>T1DE.r‹1>@1,;b>@.BN

+@(!1+@>TF‚*3GBUhs*_+TF[.B3 lš+T(13/21:(*r‹1*H3FsDE14/`*u._bR>T1X>@1KM13P+@F&3GH+@(1 KM(*X1._b +T(14.BtAC1DE+5#`‘Y./

+@.k,$.H+T(F‚K2FsKY,$1K@DE>TF[t1,\F&3 +T(1:KM1D‰+@F[.J3#!#

2(1:IY1hs*_U3*u >TF‚*3GBUhs*_+TF[.B3S*3,OF[+QK ,$U*hl$+T(1R.B>T.B3.F,$F‚*GB>g*_N‡l!*_>@1“KMUtAC1D‰+QK .b?N^*_AC.B>

F[3P+@15>@1KC+F[3 €R.BNOXU!+Q*_+TF[.J3*_h1.JNO1+@>Tu‹#‡')F[+T(>T1KMX1D‰++@.c+@(!1 F&NOXh&1N\1‰3P+Q*_+TF[.J3l+@(1>T11„$FsKM+

N*3$u *_h[GJ.B>TF[+@(!NK b.B> DE.JNOXU+TF[3G +@(1 IY1hs*_U3*u:+@>@Fs*_3GJUh‚*+@F&.B3O.b*4KM1+R.bJRu”JI X.JF[3P+QKE#Rj‡*3$u

DE.JNOXU+T1F[+ /`F[+@(c*3 .JX+@F&N*_h+@F&NO1:DE.BNkXh[1„PF[+Cu0._b PRYh&.BGR Q# 2(1\t*BKMFsDF‚,$1*\F‚KY+T.UKM1H+T(1

,$F&r$Fs,$10*_3,cDE.J3 U!1>:KM+@>g*_+T1GBu †= • …B<l y•P…Bpl .B>g… V# 2(!1KM1k*_h&GB.B>TF[+T(NK-*>@10>Q*_+T(1>:DE.JNOXh[15„

*_3,,PF^DEUh[+4+T.‡F[NOXh&1N013‹+-1 Z1DE+TF[r‹1h&u‡*3, +@(.JUGB(+@(!1u *>@1.BX!+@F[N^*_hRF[3D‰.BNOXh&1„PF[+CuBlZ+@(1u

(*r‹1.B31-N^*_F[3k,$>g*/`t*BDQˆ l‹3*NO1h[u:+@(*+`+@(15u0*_>@1 3.B+ F[3DE>T1NO13P+Q*h / 1Y(*r‹1 +@./ *_F&+ U!3$+TF[hZ*_h&h

,*+Q*-X.BF[3P+QK(*r‹12t113 DE.Bh&h[1DE+T1,-t1Eb.B>T1`/21D*30KM+Q*>@+?+T>@Fs*_3GJUh‚*+@F[3!G-+@(15N‡# 2(F‚KF‚KyF&3HDE.J3$+T>Q*JKM+

/`F[+T( /`(*_+4F‚KY3!11,$1, b.J>4.BU>:*_X!Xh[FsD*_+TF[.B3c/`(1>T10,*_+g*7*>@10DE.Jh[h[1DE+@1,SKM1 ‹U13P+@Fs*_h&h[uS*_3,‡N0UKM+

t14X>@.PDE1K@KM1, .B3S+@(1 u #

.B>@+TU3*+@1h&u‹l +@(!1>@101„PF‚KM+ KM15r‹1>Q*h?F[3D‰>@1NO153$+g*_h*h[GJ.B>@F&+@(N^KY/`(F‚DQ(c*_>T1OKCUt.JX+@F&N*_h4•V…

tU+4N*+QDQ(7.JU>3151,K -+T(1uc,$.3.J+4F[NOX.‹KC1H+@.SDE.JNOXU+T10*_G‹*F[3 +T(1H/`(.Jh[10+@>@Fs*_3GJUh‚*+@F&.B3 *_+

1*JD (HF&3KM1>T+@F&.B3# 2(1`t*BKCF‚D`F‚,$1*Y._bZ+@(!1KM1`F&3DE>@15NO13P+Q*_hP*h[GJ.B>@F&+@(N^K FsK ‹UF&+@1`KCF[NOX!h[1 KCUXX.‹KC1Y/ 1

(*r‹1R+@>TF‚*_3!GBUhs*_+@1,“+T(1?m>QKC+TRHX.BF&3$+gKElJ*_3,4+@(*+y/ 1 (*r‹1ACUKC+NO1*JKMU>T1,-+@(1BR KH%5KM+5#?' 1R1„PXh[.JF[+

+@(!1 Y._b +@(10IY15h‚*_U!3*u7+@>TF‚*3GBUhs*_+TF[.B3c/`(FsD (SF‚KY+T(1b.Bh&h[./`F[3G!#Žb

F‚K*OIY15h‚*_U!3*u +T>@Fs*_3GJh[1‹lZDE.B3KMF‚,P1>-F[+gKYD‰F[>QD‰UNK@D‰>@F[t!F[3G ,$F‚KCˆ # 2(*+4,$F‚KCˆS,$.‹1K-3.B+“DE.B3P+Q*F[3‡*3$u

.B+T(1> ,*_+g*;X.BF&3$+5# 2(F‚KRX>@.JXš15>@+CuH._b+T(1-,PF‚KMˆ!K FsK?31DE1KTK@*>@u0*_3,OKMU^DEF&13$+?b.J>2*4+@>TF‚*_3!GBUhs*_+@F&.B3

+@.t1k* I1h‚*U3*u7.J31‹#\€R.JNOF[3Gkt*BDQˆ7+T. .JU>431/ ,*_+g*^X.JF[3P+l/21H.J3h[u‡(*r‹1:+@.S,$1+T1>@NOF&31

+@(!.‹KM1:,$FsKMˆ!K .b?+T(1-1„PF‚KC+@F[3!G;+T>@Fs*_3GJUh‚*+@F[.J3F[+ b *h[hsK F[3P+@. +T(14DE.J>@>@1KMX.B3,PF[3GH+T>@F‚*3GBh&1K`*_>T1-1,$GJ1

DE.J331DE+T1, *3,H>@1+T>@F‚*3GBU!h‚*_+T1`+@(15F[>R,*+Q*-X.BF&3$+gKq+T.BGJ1+@(15>?/`F[+T(H+@(131/X.BF&3$+5# 2(1`+@>TF‚*_3!GBh[1K

/`(. KM1HD‰F[>QD‰UNK@D‰>@F[t!F[3G4,PF‚KMˆ^DE.B3P+Q*F[3K*X.BF[3P+ *_>@1A /`F&+@(+@(14X.BF&3$+5#

Ž37+@(!1;hs*BKC+fb1/’u‹1*>QK`F[3DE>T1NO13P+Q*h*_h&GB.J>@F[+T(NK2/`(FsD (S*>@13!.B37.JX+@F&N*_hF&3+@(1:/2.B>gKM+ D*BKC1

tU+ /`F[+@(c*0GB.‹.P,cDE.BNOX!h[1„PF[+Cu0(*r‹1t113SX>@.JXš. KM1,Z# 31.b2+T(1N 4•V… y/`(!.‹KM10/2.J>QKC+4D*BKC1

+@F&NO1DE.JNOXh[15„$F&+Mu4FsK! PR

RtU+2/`(!.‹KM1:*r‹1>g*_GB1 DE.JNOXh[15„$F&+Mu4FsK! PR

3 R ?(*JK2t113OUKC1,\+T.H>@1X

>@1KM13P+`KC+@1>T1.0,*_+g*$# 2(F‚KRF‚K`,$1K@DE>TF[t1,\F&37,P1+Q*F[h‚KRF[3 j `dBp #2•‹.JNO1>@1DE153$+ *_h[GJ.B>TF[+@(!NK UKC1

*0,*_+g*\KC+@>@UDE+@U>T1h&F[ˆ‹1+@(1 IY1h‚*U3*u >@11 `…B–$l +T.\U!X ,!*_+@1H>g*_XFs,$h[u0+@(!1+@>TF‚*3GBUhs*_+TF[.B3

*BK0K@*F‚,F[3}+T(1X>@15r$F&.BUKHX*_>g*_GJ>Q*_X!(l`1*BDQ(}+TF[NO1k*{3!1/ X.BF&3$+0F‚KF&3KM1>T+@1,Zl +T(17F&3DE>@15NO13P+Q*_h

*_h&GB.J>@F[+T(NK2N0UKM+ h[.PD*+@1:+@(14+@>TF‚*3GBh&1KfF&3SDE.B3 F‚DE+/`F[+T( +T(F‚K`X.BF&3$+5# 2(1:X>@F&3DEF&Xh[1-.bR+@(!1HIY1

h‚*U3*u >@11 F‚K2+@(!1“b.Jh[h&._/`F&3G F[3KC+@1*J,O._b 1h[F&NOF[3*+@F&3G4+@>@Fs*_3GJh[1K`,PU>@F&3GH+@(1“,$FZ1>T13$+ KM+T1XK2._b

KM+T1X}>T1h‚*+@F&.B3KC(F[XKt1+C/ 1131*BDQ(+@>TF‚*_3!GBh[1K._b4+T(1‡KCUDDE1KTKMF&r‹1IY1hs*_U3*u >@Fs*_3GJUh‚*+@F&.B3’*_>T1

N*F[3P+Q*_F&31, #?')(!13‡*HX.BF[3P+F‚K`F[3KC1>@+T1, l*_h&h+@(1 IY1hs*_U3*uO+T>@Fs*_3GJh[1K`F[3SDE.J3 F‚D‰+-*>@1H,P1DEh‚*>@1,

,$1*J, ˆ$F&h[h&1,t$u:+@(1X.BF&3$+ Ql$t!U+R+@(1u0KC+@F&h[h1„PF‚KC+?F[30+@(!1 >@11 # 2(131/ +@>@Fs*_3GJh[1K?t1DE.JNO1KM.J3K

._b +@(1KM1,$1*J,”+T>@Fs*_3GJh[1K‰# 2(1 IY1hs*_U3*u:+@>@Fs*_3GJUh‚*+@F&.B30F‚K b.J>@NO1,tPu+T(1`U3F&.B30._b *_h[h!+@(1*h[F[r‹1

+@>TF‚*3GBh&1KE# 2(1 KM+@>TUDE+TU>@14._b +T(1HIY15h‚*_U!3*u >T11HKCX11,K`UX‡+@(!1h[.PD5*_h[FsK@*+@F[.J37.bR+@(1:+@>TF‚*_3!GBh[1K

F[3DE.B3 F‚DE+:/`F[+@(*7Xš.JF[3P++T.‡tš1kF[3KC1>@+T1, #‡e`1DE13P+@h&u‹l*3}*h[GJ.B>@F&+@(N t*JKM1, .B3+T(1IY1hs*_U3*u

>@114(*JK2tš15137X>T.BX.‹KC1, tPu †I-j /`(. KM1:>Q*_3,$.BNOF 1,0DE.BNkXh[1„PF[+Cu:F‚K h&.BG R RX1>YU!X ,!*_+@1 #

2(FsK *h[GJ.B>@F&+@(N *h[h&._/K?t.B+T(O+@(!1YF&3KM1>T+@F&.B3k*_3,H+@(1 ,$1h&1+@F&.B3H.bXš.JF[3P+QK‰l /`(!F‚DQ(HF‚K?1„!*BDE+Th[u/`(*_+

/213151, +@.cUXZ,*+@1‡*7r‹.Jh[UNk1\>T1X>@1KM13P+Q*+@F[.J3._b +@(1Ob>@1517KCX*BD‰1^b>T.BN *j”JI N*XN*B,$10._b

KM15GBNO13P+QK‰lP*BK`1„$X!h‚*_F&31,\F&3SKM1DE+TF[.J3 !#1

bU>@+T(1>X>@.JXš15>@+CuO._bq+@(1:IY1hs*_U3*u+T>@F‚*3GBU!h‚*_+TF[.J37/`(FsD (F‚K2r‹15>@u0>@1h&1rJ*_3P+`+@.0.BU> *_XX!h[F

D*+@F[.J3F‚K2>@15h‚*_+T1,\+T.H+@(1:KMˆ‹15h[1+T.B3# Ž 3,$11, lPFb/ 14DE.B3KMF‚,P1>`+@(1:DE153$+T1>QK?._bq+@(14I1h‚*U3*u,$FsKMˆ!K

*_3,“AC.JF[30+@(. KM1 DE13P+@1>gKy/`(.‹KC1Y+T>@Fs*_3GJh[1K?KC(*_>T1-*3\1,$GB1tU+ ,$.43.J+2D‰>@.‹KTK?+@(1t.BU3,!*_>@u0._bZb>@11

KMX*BDE1‹l/ 14.Bt+g*_F[3 *\KC1+._bRX.Bh&u$GJ.B3*hyh&F[31K2+@(*+-*>@1:*\KCUtKC1+Y.bR+@(!1 R.J>@.B3!.

F

,$F‚*GB>Q*Nv._bR+T(1

KM15+-._bRNk1*BKCU>@1,X.BF[3P+QK j `dJp #R.B>@.J3. F ,$Fs*_GB>g*_NK(*r‹1t113S(1*r$F&h[uOUKM1,7F&37>@.Jt.B+@FsDK

*BK*0KMUXX.J>@+Yb.J>Y+T>Q*_AC1D‰+@.B>TuOXhs*_33F&3Gk*_3,7+T(1-b *BDE++@(*+Y+T(1 R.J>@.B3!. F ,$F‚*GB>Q*Nv._b?b>T11:KMX*JDE1

F‚K2X!>@1KC13$+`F&3.BU>`>T1X>@1KM13P+Q*+@F[.J3SDE.BNO1K2*BK*HKM+T>@.J3GKMU!XX.B>@+`b.B>.BU!>*_XX>T.‹*JD (#

K R H

!24%5&( = 0^*\% 918 9 6307-O%'0 ´@( = *307%5³4-O630L(.*32M%

Ž3.BU>0D*JKM1/21(*r‹1ONk.B>@10+@(*_3}X.JF[3P+QKElR/21k(*r‹1KM1GJNO13P+QK‰# Žb /21ACUKM+0+@>@Fs*_3GJUh‚*+@1+@(!1F[>

13,PXš.JF[3P+QK`*JK`KM(./`3F[3\mGBU>T1d/ 1-.Bt!+Q*_F&3+@(14KC. D*_h&h[1,0U3DE.J3KM+T>Q*_F&31,O+@>TF‚*3GBUhs*_+TF[.B3.b?+T(1

KM15+-._bfKM1GJNO13P+QK`._bRmGBU!>@1k…$#HŽ+4D*3‡t1OKC113‡+@(*_+:KM.JNO1HKC1GBNO153$+gKY*>@1H3.J+I1h‚*U3*u71,PGB1K‰#

2(FsK F‚K *;X>T.Bth&1N tš1D*_UKC1-+@(!1Y315„$+ KM+@15X,$1KTDE>@F&tš1,”F&3O+@(1`b.Bh&h[./`F[3!GHKM1DE+@F&.B3ON*u:X>@.P,$UDE14*

h[1K@K`*BD5DEU>Q*+@1:>@1X>T1KM153$+g*_+@F&.B3._b b>@11:KMX*BDE1:+@(*37Fb *_h[hKC+@1>T1.0KM15GBNO13P+QK`*>@1 IY1h‚*U3*uO1,PGB1K‰#

2(1OX!>@.Bt!h[1N D*3tš1k1h[F&NOF[3*+@1,t$uc*J,,$F&3GcKM.JNO1\Nk.B>@10X.BF&3$+gK;.B3 KC.BNO10._b+T(1KM15GBNO13P+QK

+@(!1 >T1KMU!h[+QK.J3+@(1SK@*NO1KTDE131*>@1 KM(./`3F[3 mGJU>@1Sd‡/`(F‚DQ( F‚K+T(1SKM. D*h[h&1,DE.B3KC+@>g*_F[3!1,

IY1hs*_U3*u4+T>@F‚*3GBU!h‚*_+TF[.J3._b+@(1 KM1+.bZKM+T1>@1. KM15GBNO13P+QK‰#Ž3 j `dBp ‹+@(1 *_U+T(.B>gK?,$1r‹1h&.BXX1,

*_3^*_h[GJ.B>TF[+@(!N +T(*_+ DE.B3KC+@>g*_F[3k+@(1 KM1GJNO13P+QK‰# 2(!F‚K2*h[GJ.B>@F&+@(N X1>Mb.J>@NKF[3 PR

?F&3\+T(1Y/ .B>gKM+

D*JKM1‹l?tU!+H.BU>HF&NOXh[15N\13P+Q*+@F&.B3‡UKC1K OKC.‡+@(1^*r‹15>Q*_GJ1O>@U!33F[3!G‡+@F[Nk1\FsK

‹UF[+T1Yh&1K@K?+@(*_3+@(*+# 2(1t*JKMF‚DF‚,$1*4F‚K?+@.:NO.P,$FnbuH+@(1F[3X!U+`,*+Q*4t$uk*B,,PF[3G:NO.B>T1`Xš.JF[3P+QKR+@.

+@(!1F[3!F[+@Fs*_h KM1GJNO13P+QK‰# 2(!F‚K-D*3‡t10,$.B31HUKMF[3!GO+@(14bU3,*NO13P+Q*hyX>T.BX1>T+Mu.b2+T(10IY1hs*_U3*u

>@F‚*3GBU!h‚*_+TF[.J3,$1KTDE>@F&tš1,{1*>@h&F[1>4F[3c+@(!F‚K4KM1D‰+@F[.J3# 2(FsK;X>T.BX1>@+CucF[3Pr‹.Bh&r‹1Kf+T(*_+:F b +@(1D‰F[>QD‰h[1

,$1Em31, tPuc*7KC1GBNO153$+ *BKF&+QK4,$F‚*NO1+T1>,P.‹1K3.B+:F[3P+@1>gKM1D‰+4*_3Pu .J+@(15>HKM15GBNO13P+lZ+@(13 +@(FsK

KM15GBNO13P+?/`F&h[ht1 *,$1hs*_U3*uH1,$GB1._b+T(1Y>T1KMU!h[+@F&3G4+@>TF‚*_3!GBUhs*_+@F&.B3#RŽ+RF‚KR*4KMU DEF&13P+RDE.B3,$F[+TF[.B3 #

1+ tš1cKCUDQ(’* DEF[>gDEh[1S*+@+Q*JD (!1,}+@. +T(1cKM15GBNO13P+ ‹l-*3, 888

t1‡+T(1cKM1GJNO13P+QK

F[3P+@15>QKM1DE+@F&3G # Ž+HF‚KX.‹KTKMF&th[1+@. KMX!h[F[+ F&3$+T.c* m3!F[+@1k3$UN0t1>.b4KCUtKC1GBNO153$+gKKMUD (+@(*_+

3.J31O._b+T(1kDEF[>gDEh[1K *_+@+g*BDQ(1,‡+@.S+@(. KM1KC1GBNO153$+gKYF[3P+@1>gKM1D‰+QK-*3$uc._b+T(1 KE#‘1>@1H/ 1\.JNOF[+

+@(!1O,$15+Q*_F&h‚K-._b+T(F‚K4X>@15X>@.PDE1K@KMF&3G‡KC+@1X*_3, ACUKM+X!>@1KC13$+4+T(1O*h[GJ.B>@F&+@(Nob.B>:DE.JNOXU+TF[3G+T(1

DE.J3KM+T>Q*_F&31,IY1hs*_U3*u >TF‚*_3!GBUhs*_+@F&.B3

.B>4*3$uS1,$GB1HF&3

l FnbRF[+“,$.‹1K-3.B+“K@*_+TF‚Kbu

+@(10bU3,*NO13P+Q*_hRX>T.BX1>T+Mu lyX>T1X>@.PD‰1K@KF&+H*_3,*J,,

+@(1:DE.J>@>T1KMX.B3,$F[3G031/ Xš.JF[3P+QK2F&37*Hh[FsKM+5#

€R.BNOX!U+@10+@(1kIY1hs*_U3*u >@Fs*_3GJUh‚*+@F&.B3 .b+@(1031/

KM1+`.bRX.BF&3$+gKE#

yF&GBU>T1… …”BI]N^*_XO.b *\KTDE131

' 10KT*/]F&3cKM1D‰+@F[.J31P#1$#1\+@(*_+-/`(153‡+M/ .O.B> NO.B>T1HKM15GBNO13P+QK`*>@1HNO15>@GB1, l/21HˆP3./b>@.BN

/`(1>T1H1*BDQ(7X*_>@+4.b2+T(1H>@1KMUh&+@F[3!GOKC1GBNO153$+(*BK-t113cKC113 KM151mGBU>T1OV g#;')(153‡Xš.JF[3P+QK4*_>T1

*B,!,$1, .J3cKM1GJNO13P+QK`,$U!>@F[3!G\+T(1HDE.J3KM+T>Q*F[3F&3G\X!(*BKC1‹l +@(!1uOF[3!(1>@F&+Y+T(14r$FsKMF&tF[h&F[+CuO*+@+T>@F[t!U+@14._b

+@(!1X*>@+-.b2+T(1HKC1GBNk13$+ +@(15u (*r‹1:t1137X!U+-.B3 l *JK-KM(./`3‡F&3 mGJU>@1%5p$# 2(!F‚KYF&3$b.B>TN*_+TF[.B3

.B3 +@(10X.BF[3P+QK4F‚K-r‹1>Tu‡F[NOX.B>T+Q*3$+Yb.J>+T(1\X!>@.PDE1KTK;/`(FsD ( DE.JNOXU+T1H+@(1Hb>@11kKMX*BDE1\b>T.BNo+T(1

+@>TF‚*3GBUhs*_+T1,ON*_X._b +@(!1D‰.B3KC+@>Q*F[31,7KM15GBNO13P+QK‰lP*_3,/`F[h[hZt1,P1K@DE>TF[t1,OF[3S,$15+Q*_F&h‚KRF[3SKM1D‰+@F[.J3

#†#

Some segments (bold) are not edges

of triangles (thin) All Segments are edges of triangles.

Some points have been added

yF&GBU>T10d 4WY3D‰.B3KC+@>Q*F[31,‡*_3,cDE.B3KM+@>g*_F&31,‡IY1hs*_U3*u‡+T>@Fs*_3GJUh‚*+@F[.J3‡._b +@(10KC1+4._b KM1GJNO13P+QK

._b mGBU>T1:…

seen by R1

Seen by R1 and R3 Seen by R3

R1 R2

R3

Seen by R2 and R3

Seen by R2

= constrained points

yF&GBU>T1 %5p 7')(!13)*cKM1GJNO13P+FsKHDE.B3KC+@>g*_F[3!1, l2+T(17rPF‚KCF[tF&h[F[+Cuc*+@+@>TF[tU!+@1F‚KF&3(1>TF[+@1,tPu +T(1

*B,!,$1, X.JF[3P+QK

K R W Y 01(+*\%'³ ´O%50 * 0^6\6,´ 0^% *3&(+*\%'³( *30^%'³4-6301(+*324%

')(13 *”3!1/9KM15GBNO13P+2FsK bUKM1, /`F[+T(‡*37.Bhs, .J31‹l+@(!1.Bhs,7KC1GBNO153$+ F‚K`1>Q*JKM1,\b>@.JNv+@(1:+@>@Fs*_3

GBU!h‚*_+TF[.J3l*_3,+@(14315/ .J31N0UKM+ t1;F&3KM15>@+@1, #

Y1/ KM1GJNO13P+QKR+@(*+`(*r‹1;3!.B+tš1513SKM113t1Eb.J>@14N\UKM+2t14F[3KM1>T+@1,O+@.‹.!#

2(1kKM1GJNO13P+QKY.b`+@(!1\UXZ,*+@1,‡N*X‡N\UKC+-t1OD‰.B3KC+@>Q*F[31,{t1Eb.J>@10tš15F[3GF[3KM1>T+@1,7F&3 +T(1

+@>TF‚*3GBUhs*_+TF[.B3 # 2(1S*_h&GB.J>@F[+T(N /`(F‚DQ(DE.J3KM+T>Q*_F&3 *‡KM1+H.b4KC1GBNO153$+4F‚K43.B+0F[3DE>@1Nk13$+g*_h KC11

KM1DE+@F&.B3O!#[% g# .J>@+@U!3*_+T1h[u‹l‹/ 1;3!.B+@FsDE1,\+T(*_+`/`(!13O/214NO.r‹1KC.BNO1 KM1GJNO13P+QK?b>@.BN *HKM15+`._bq*_h

>@1*B,$u4D‰.B3KC+@>Q*F[31,KM15GBNO13P+QK‰l.B3h&u-+@(!12KC1GBNk13$+gKšF&3;+T(123!1F[GJ($t.B>T(.‹.P,311,“+T.Yt1 DE.B3KC+@>g*_F[3!1,

*_G *_F&3#

.:KCUNLUX lB+@(1KC1GBNk13$+gK +@(*+RN\UKC+t1Y>T1NO.r‹1,fb>T.BNL+T(1`+@>TF‚*3GBUhs*_+TF[.B30*>@1`+T(.‹KC1`/`(F‚DQ(

*_>T1-bUKM1,O/`F&+@(SKM.JNO1->@1DE13P+@h[uH.JtKM15>@r‹1,OKC1GBNk13$+gKElP*_3, +@(!.‹KM14/`(FsD (3151,\+@.0t1:DE.B3KC+@>g*_F[3!1,

,$F 15>@13P+@h&u-,$U!1 +T.YNk.P,$FmZD5*_+@F&.B34._b+@(1F&> 315F[GB(Pt.B>T(.‹.P, /`(.‹KC1213,$X.BF[3P+QK >T1N*F[31,+@(1 K@*NO1‹l

+@(!.BUGJ(Q#

Update the 2D map

Constrain the updated map

Delaunay Triangulation Remove segments fused

Add new segments constrained, update segments constrained differently than before.

Updated map

F[GBU!>@10%B% 2WYXZ,*+@F&3G\+T(1:,$1hs*_U3*uO+T>@Fs*_3GJUh‚*+@F[.J3

K R K ¯ 0 *\%'³ Y5( ´( *307%'³M6 & 2 Y -/(.*\%5³N( A & YO0

‘-*rPF[3G DE.J3KM+T>@UDE+T1,)+@(1 IY1hs*_U3*u)+T>@F‚*3GBU!h‚*_+TF[.J3 .b0.BU>N*XlY/ 1 3./o/`FsKM(’+@.}Fs,$13P+@Fnbu

+@(!.‹KM10+@>TF‚*3GBh&1K-/`(F‚DQ(c*_>T1HX*_>T+QK;.b`+T(1b>@110KCX*BD‰1‹#0Ž 3‡.J>Q,$15>;+T.S,$.+@(FsKEl/21H1„PXh&.BF[+“*^r‹15>@u

KMF&NOXh&1-r$FsKMF&tF[h&F[+Cu\X!>@.BX1>T+Cu7/`(FsD (SF‚K`tš1KM+-1„PXhs*_F[3!1,Ot$uh[.‹.Jˆ$F&3G*+YmGBU!>@1%”P#4Ž3‡+@(FsK mGJU>@1 l

+@(!1\X(Pu!KMF‚D5*_h KM1GJNO13P+QK‰l /`(F‚DQ(c>@1X>T1KM153$+4>@1*_hR.BtKC+Q*JDEh[1K4*>@1O+T(1\+T(F‚DQˆ7h&F[31KEl*_3,c+@(1O.J+@(15>

IY1hs*_U3*uS1,$GJ1K-*_>T1H+@(F&37h&F[31K‰#-' 1:3.B+TF‚DE1H+T(*_+“KM.BNk1HKM15GBNO13P+QK`/`(FsD ( D*_3‡t10KM1513‡t$u+T(1

>@.Jt.B+2F&31BI’KMX*JDE1*_>T1Y3!._/ t1(F[3,HKM.BNk12.B+T(1> KM15GBNO13P+QK/`(FsD (H>T1X>@1KM13P+y.JtKM+g*BDEh&1K?t1+C/2113

+@(!1>@.Jt.B+:*_3,7+T(.‹KC1HKC1GBNO153$+gK2tš15(F[3,+@(15N‡#`Ž37+T(1;mGJU>@1 l +@(1:X.BF[3P+ (*JKYt113 KM113St$u

+@(!1H>@.Bt.J+ltU+-+T(1H.BX!+@F‚D5*_h>Q*u7FsKYF[3P+@1>gKM1D‰+@1,\m>gKM+-tPu7+T(10KM1GJNO13P+ )*3,7+@(!10KM1GJNO13P+ ‰#

2(. KM14+@>TF‚*_3!GBh[1K`DE>@. K@KM1,\t$u0+@(!1;.JX+@FsD*_hZ>Q*u0tš1‰b.B>@14F&+2NO115+QK?+@(1m>QKC+KM1GJNO13P+ *>@14N*_>Tˆ‹1,

*BK2b>@1514KMX*JDE1‹#

P

Robot position S Segments

Delaunay edges

S1

yF[GJU>@10%y” `*BKMFsDYX>TF[3DEF&Xh[1 ._bRUKCF[3GH+T(14r$FsKMF&tF[h&F[+CuHX>@.JX1>@+CuO+T.HN*_>Tˆ\1NkX+CuH+@>@Fs*_3GJh[1K‰#

' 1KT*/)F[30KM1DE+@F&.B31$#1$#1“+T(*_+R+T(1KM1GJNO13P+QK._b .BU>~”BI)N*XHN*u;(*r‹1`t11530KM1130X*>@+@Fs*_h&h[u

b>@.JN KC1r‹1>g*_hr$F&1/`X.BF&3$+gKElPKM.H+T(1YX.JF[3P+QK2.B3k+@(1KC14DE.J3KM+T>Q*_F&31,OKC1GBNk13$+gK?N*u(*r‹1tš1513KM1513

b>@.JN KC1r‹1>g*_hRr$F&1/`X.BF&3$+gK;+@.‹.!# •‹UX!Xš. KM1+@(*_+H+@(!1O1„!*_NOXh&1HN*_Xc._b+T(1\mGJU>@1 %17(*JKtš1513

UXZ,*+@1,OKC1r‹1>g*_h+TF[NO1Kt$uHNO15>@GBF&3GM”JIKM1GJNO13P+QK?KC113Hb>@.JN KM1r‹1>g*_hrPF[15/`Xš.JF[3P+QKR*_3,\+@(*+2+T(1

X*>@+4._b +@(10KC1GBNO153$+-/`(!F‚DQ(‡DE.J3$+g*_F[3K“+T(10Xš.JF[3P+ (*JK;t113cKC113‡b>@.JN +C/2.S,$F 15>@13P+->@.Jt.B+

X.‹KCF[+@F&.B3K -*3, IE# Ž +K231DE1K@KT*_>TuO+@.@hs*_U3DQ(H+C/2.H.BX!+@F‚D5*_h>Q*u!KEl ƒ*3, I]+@.0N*>@ˆ

*_h&h+@(141NkX+CuH+@>@Fs*_3GJh[1K‰#

.kKMU!N UXl+T(1:*_h[GJ.B>TF[+@(!N b.B>N*_>Tˆ$F&3G+T(141NOX+CuH+T>@F‚*3GBh&1K2F‚K2+T(1-b.Bh&h[./`F[3G

P

R1

S Segments

Delaunay edges

S1

Optical Ray

R2

F[GBU!>@10%1 2W-KCF[3GH+T(14r$F&1/`X.BF&3$+gK2+@.HN*>@ˆ01NOX+CuH+@>TF‚*3GBh&1KE#

‹9 [ K$ T



p&* 58F2h?s&%S9*82&>H4*"$2qQ€

p&Q* [#:WI$"$2"!

?wC 'X&f&

$2# *#2q"2

$

p&Gs* (S9*82&>4$&%h !W5"R#W=R&" *4;*=

!

b„‡F2Q&%;]?AGC'!

( ‰~€

)*

*)+%U*GO?A5"* !r

)

+‡?&>?% &H&>'B 5d&

;*=,!

h !CMOH&Ž9X*S>CM2V&2

"5Kr!49S6 !G2W"$2H&J

3DE14+@(14+T>@F‚*3GBh&1K2t1h[.J3GBF&3G0+@.H+@(!1“b>T11:KMX*JDE1:*_>T1;N^*_>@ˆ‹1, l‹F[+`F‚K2r‹15>@u01*BKCu +T.0DE.BNkXU+@1

*\(PUh[h.bR1*JD (S.BtKC+Q*BD‰h[1tPuSDE.B3!31DE+TF[3Gk*_h&h,$1h‚*U3*u1,$GJ1Kf/`(!F‚DQ(7t1h[.J3GO+T.\t.B+T(c*_3‡1NkX+Cu

+@>TF‚*3GBh&1-*3,\*-+T>@F‚*3GBh&1Y/`(!F‚DQ(H(*BKR3.B+Rt1130N*_>Tˆ‹1, # `+?+@(FsK?Xš.JF[3P+l‹/21`ˆP3./}+T(1`b>@11KCX*BDE1 l

+@(!1\r‹.J>@.B3!.BF S,$F‚*GB>g*_N KM151KM1D‰+@F[.J3 !#[%Qly+T(10Xš. KMF&+@F[.J3 *_3, KC(*_X1 *BK:DE.B3Pr‹1„‡X.Bh&u$GJ.B3K -._b

+@(!1;.JtKM+g*BDEh&1KE# 2(!1>@.Jt.B+ D*_3S3./ Xhs*_3S+@.HNO.r‹1 K@*b15h[uO+T.O*315/ X.‹KCF[+TF[.B3 #

x3u

-DE+TU*_h&h[u‹l/21:(*r‹1HF[NOX!h[1N013‹+@1,O*,$1NO.0F[3‡/`(FsD (S.BU>4>T.Bt.B+-m>gKM+4F[3P+@1GJ>Q*+@1K2mr‹1HrPF[1/K`t$u

>@.J+Q*+@F[3!G .B3h&u+@(1S+@>@F&Xh[15+H._b:D*NO1>g*BK0*_3,}tU!F[h‚,!KH* X*_3!.B>Q*NOF‚DON*X}.b4+@(1b>@11 KMX*JDE17F&3

b>@.J3$+:._b (F[N l +T(13 X!h‚*_3KH*‡K@*b1+T>Q*_AC1D‰+@.B>Tu‹lyNk._r‹1K-*_3, +g*_ˆ‹1K:*_G‹*F[3 *7X*3.B>g*_NOFsDBl?*3,KM.

.B3 #[#[# 2(!F‚K t1(*rPF[.J>`F‚K DEh[. KM1-+T.H+@(14(PUN*_30.B3!1 ?/214h[.‹.Jˆ*_>T.BU3,^t1Eb.J>@14NO.rPF[3G4F[3^*(.‹KC+@F[h&1

13Pr$F&>@.J3NO13P+#

2(1:*_h&GB.B>TF[+T(N

%B# *_ˆ‹1 *YX*_3!.B>Q*NOF‚DRr$F&1/c._bZ/`(*_+RFsK F&3;b>@.J3$+ ._b+T(1`>@.Jtš.J+l‹UXZ,*+@1`+@(!1%”BI}N^*_XlJDE.BNkXU+@1

+T(1-b>@114KCX*BD‰1‹#

”$# .‹.Jˆ}Fb +@(!1>@1S*_>T1 $ 9H+@. *_XX!>@.‹*JD ( +@(1SGB. *_h# Ž b“KMUD ( X*BKTK@*_GJ1K\1„PF‚KC+l

DE.JNOXU+T1O+@(!17X*+@(KH+@.cGB. +T. +T(1N‡#Žb +@(15>@1S*_>T17NO.J>@1+@(*3}.J317X.‹KTKMF&th[1SX*BKTK@*_GJ1‹l

DQ(.‹.‹KC1+T(14tš1KM+`.B3!1H*_Nk.B3GH+T(1N‡#

1$#Žb+@(!1>@1F‚KH*7X*BK@KT*_GJ1‹l2X1>Mb.J>@N +T(1SDE.B>T>@1KCX.B3,$F&3GcNO.r‹1N\153‹+QK4DE.BNkXU+@1,{F&3}KC+@1Xb”P#

*3, GJ.\+T.0KM+@15X %B#

#Žb 3.X.‹K@KCF[th&1\X*BK@KT*_GJ1 (*BK-t113‡b.BU3,Zl X1>Cb.B>TN * KT*b10NO.r‹1NO1‰3$+`F&3‡+@(1Hb>@1510KMX*JDE1

KC.\+T(*_+-+T(1431„P+YX*_3.J>Q*_NkF‚D-r$F&1/’/`F[h[hZ>@1r‹1*_h+@(!F[3G Kf+T(*_+(*r‹13.J+Yt1153‡KM1513tš1‰b.B>@1 #

.0+@.0KM+T1X %J#

?„$Xhs*_3*+@F&.B37.bRKC+@1Xm”

Y31,$GJ1;.bR+@(!1;+T>@Fs*_3GJUh‚*+@F[.J3F‚K`* $ 9RFb

Ž+`F‚K2.J37+T(14tš.JU3,*>@u._bq+@(1-b>@1514KMX*JDE1‹#

Ž+?FsK *3\1,$GB1._b +M/ .4+@>@Fs*_3GJh[1K .B3!1fb>T.BNƒ+@(!12b>@11KCX*BDE1 *_3,\.B31F[3P+@15>@3*h$+T.+@(!1DE.B3Pr‹1„

(PUh[h Q#

Ž+`F‚K23!.B+ *X($u!KMFsD*_h KM1GJNO13P+#

Ž+`F‚K2h&.B3GH153.BUGJ(# 2(1:>@.Bt.J+YN0UKM+ t1*th[14+T.ODE>T.‹KTKfF&+Q#

2(F‚K,P1Em3F&+@F[.J34NO1*_3Kš+@(*_+ *2Xš. K@KCF[th&1 X*BK@KT*_GJ1`F‚K*_3:1,$GJ1?._b+T(1 DE.B3Pr‹1„`(PUh[hJ+@(*+y(*BKtš1513

tUF&h[+`tPuO+@(!1;X!>@.PDE1KTKf+T(*_+>@15NO.r‹1K?+@(141NOX!+MuH+T>@Fs*_3GJh[1K‰ltU+,$.‹1K`3.J+Y>T1X>@1KM13P+KM.JNO1+@(!F[3G

+@(*_+-(*JKf>T1*_h&h[uOt1153cKM113 KM114mGBU>T1O%‰ Q# Ž37+@(*_+ D*BKC1‹lš+T(1>T.Bt.B+4FsK`*_+@+T>Q*JDE+@1,7tPuKCUD ( *_3

@U!3ˆ$3!._/`3 “X*_>@+R.b +T(1`N*X(1`tU!F[h[+ KM.-b *>#?‘1`/`*_3P+QK+@.4KC11`/`(*_+Kyh&u$F&3G-+T(1>@1 *_3,HFb+@(1>T1

F‚K:*7X*BKTK@*GB1+@(*+0D*_3 h[1*J, (F[No+T.‡+@(1kGB.‹*h#

b DE.BU!>QKM1k+@(1>T1ON*u715„$FsKM+HKC1r‹1>g*_hRX.‹K@KCF[th&1

X*JK@K@*GB1K‰lKM.JNO14N*u\3!.B+tš1:>@1*JD (*_th&1;FnbR.B3S+@(1:/`*uO+T.\+T(1N +T(1>@1 *_>T13*>@>@./G *_XKN*B,$1

tPuH.BtKC+Q*BD‰h[1K2.J>2Fb +@(1 >@.Bt.J+KM(.JUh‚,OGB.H.JU+2.b+@(1b>@11 KMX*JDE1 KM11YmGBU>T1H%_V Q#`•‹./`(*+`/21 ,$.

F‚KR+@(*+`/21Ym>QKM+,P1+@1>TNOF[3!1`+@(1-X. K@KMF&th[14X*JK@KT*_GB1KEl+@(13^DE.BNOX!U+@1-+T(14KC(.B>T+@1KC+`/`*uKR+@.H>T1*BDQ(

+@(!1N *_3,KC.BNO1+TF[NO1Ky+@(15>@1-FsK23.H/`*u0+@.HGB.0+@.0*GJF[r‹13kX*BKTK@*_GJ1H*JK21„$X!h‚*_F&31,\1*_>@h&F[1> Ql‹+@(!13

DQ(.‹.‹KC1H*_Nk.B3GH+T(1:,$FZ1>T13$+ /`*u!K2+@(14.B3!1+@(*_+N*_ˆ‹1+@(14>T.Bt.B+GB.kDEh[. KM1>`+T.\+T(14GB. *_h#